TABLES 

COMPLEX  HYPERBOLIC  AND 
CIRCULAR  FUNCTIONS 


KENNELLY 


UNIVERSITY  OF  CALIFORNIA 
AT   LOS  ANGELES 


6<TU£_puPuAA*--'' 


.^.  T 

ii- 


TABLES 

OF 

COMPLEX  HYPERBOLIC  AND 
CIRCULAR  FUNCTIONS 


BY 
A.  E.  KENNELLY,  Sc.D.,  A.M. 

PROFESSOR  OF  ELECTRICAL  ENGINEERING  IN  HARVARD  UNIVERSITY 


SECOND  EDITION 

REVISED  AND   ENLARGED 


CAMBRIDGE 
HARVARD  UNIVERSITY  PRESS 

LONDON:  HUMPHREY  MILFORD 

OXFORD  UNIVERSITY  PRESS 
1921 


First  edition,  March,  1914 
Second  edition,  February,  1021 


Library 


PREFACE 

THE  tables  in  this  book  present  hyperbolic  and  circular  functions  of  a  complex 
variable,  both  in  polar  and  rectangular  coordinates.  Such  complex  functions 
have  not  hitherto  been  published,  except  over  a  very  restricted  range.  They  have 
important  applications  in  electrical  engineering.  For  instance,  it  is  possible  with 
their  help  to  find  in  a  few  minutes  the  potential,  current  and  power,  at  any  point 
of  an  alternating-current  line-conductor  of  known  constants  and  terminal  condi- 
tions; whereas  the  same  problem,  to  a  like  degree  of  precision,  without  aid  from 
these  functions,  and  by  older  methods,  would  probably  occupy  hours  of  labor  and 
cover  several  sheets  of  computing-paper. 

Although  the  principal  application  of  these  functions  at  the  present  time  is  in 
dealing  with  alternating-current  lines,  especially  those  of  either  great  length  or 
high  frequency;  yet  it  seems  likely  that  other  uses  will  develop  for  them. 

The  author  desires  to  acknowledge  his  indebtedness,  for  suggestions  and  help, 
to  a  number  of  workers,  both  in  mathematical  and  practical  fields;  and  particu- 
larly to  Messrs.  C.  L.  Bouton,  W.  Duddell,  E.  V.  Huntington,  F.  B.  Jewett,  John 
Perry,  H.  J.  Ryan,  and  E.  B.  Wilson. 

A.  E.  K. 

HARVARD  UNIVERSITY 
January,  1914. 

I 

PREFACE  TO  THE  SECOND  EDITION 

i  IN  preparing  the  second  edition  of  this  book,  six  new  tables  have  been  computed. 

J  These  are  actually  extensions  of  the  tables  I  to  VI  already  incorporated.  It  has 
been  considered  advisable  to  add  the  new  material  in  new  tables  at  the  end  of  the 
volume  rather  than  to  recast  the  original  tables  in  such  a  manner  as  to  include  the 
new  matter.  The  new  matter  has  been  found  necessary  in  certain  departments  of 
electrical  engineering  to  which  complex  hyperbolic  functions  may  be  advanta- 

geously applied. 

A.  B.  K. 

HARVARD  UNIVERSITY 
June,  1920. 


442282 


TABLE  OF  CONTENTS 


PAGE 

I.    HYPERBOLIC  SINES,     sinh  (p  /d)  =  r  /_y  .    .    .   5  45°  to  90°   ......  2 

H.    HYPERBOLIC  COSINES,   cosh  (P  /5)  =  r  fa    .      "  "     "    "     ......  8 

HI.    HYPERBOLIC  TANGENTS,    tanh  (P  /S)  =  r  fy_       "  "     "     "     ......  14 

IV.    CORRECTING  FACTOR.    ^^   .                          "  "     "     "  20 

a 

V.    CORRECTING  FACTOR.    ^-?  .......      "  "     "     "  26 

(7 

VL.    FUNCTIONS  OF  SEMI-IMAGINARIES.  /  (p  745°)   .............  32 

VII.    HYPERBOLIC  SINES,   sinh  (x  +  iq)  =  u  +  w     .............  42 

VIII.    HYPERBOLIC  COSINES,   cosh  (x  +  iq)  =  u  +  iv    ............  58 

IX.    HYPERBOLIC  TANGENTS,   tanh  (x  +  iq)  =  «  +  iv    ...........  74 

X.   HYPERBOLIC  SINES,   sinh  (x  +  iq)  =  r  /_y_     ..............  90 

XL   HYPERBOLIC  COSINES,   cosh  (x  +  iq)  =  r  /_y_    .............  106 

XII.    HYPERBOLIC  TANGENTS,   tanh  (*  +  »</)  =  r  /_y_    ............  122 

XIII.    FUNCTIONS  OF  4  +  iq.           f(4+iq)=u+iv     .....    .....    .  138 

f(4+iq)  =r/7     ............  139 


XIV.    SEMI-EXPONENTIALS.    —  and  logio  (~ 


XV.    REAL  HYPERBOLIC  FUNCTIONS.    /  (x  +  i  o)  =  u  +  i  o  .........  144 

XVI.    SUBDIVISIONS  OF  A  DEGREE    ....................  150 

EXPLANATORY  TEXT  ........................  151 

XVH.    HYPERBOLIC  SINES.   sinh(p/6)  =  r  /r  5  o°  to  45°     .......  214 

XVIII.    HYPERBOLIC  COSINES.   cosh(p/5)  =  r/r        ""   "     "      .......  216 

XIX.    HYPERBOLIC  TANGENTS.   tanh(p/«)  =  r  [y_    ""   "     "      .......  218 

XX.    CORRECTING  FACTOR.    s-  "  "   "    "      .......  220 


XXI.    CORRECTING  FACTOR.  "  "   "     "      .......  222 

a 

XXII.    FUNCTIONS  OF  SEMI-IMAGINARIES.  /  (P  /45°)     .   .   .   .   .......  224 

XXIII.    HYPERBOLIC  FUNCTION  FORMULAS  .................  226 


TABLES    OF    COMPLEX    HYPERBOLIC 
AND    CIRCULAR    FUNCTIONS 


TABLE  I.    HYPERBOLIC  SINES,     sinh  (p  /S)  =  r 


O.I 


o-3 


0.4 


45 
46 
47 
48 
49 

O.IOOOO 

0.099993 
0.099987 
0.099981 
0.099975 

45.096 
46.095 

47-095 
48.094 
49.094 

O.2OOOO 
0.19995 
O.I9990 
0.19986 
0.19982 

45-383 
46.382 
47-381 
48.380 
49-378 

0.30001 
0.29985 
0.29969 
0.29954 
0.29939 

45.860 
46.858 
47-856 
48.854 
49.852 

0.40005 
0.39968 
0-39931 
0-39893 
0.39856 

46-532 
47-529 
48.526 
49.520 
50.5*3 

0.50016 
0.49944 
0.49872 
0.49799 
0.49727 

47-391 
48.388 

49-385 
50-378 
51-368 

So 
Si 

52 

53 

54 

0.099970 
0.099965 
0.099960 
0.099955 
0.099950 

50.094 

51-093 
52.092 

53-091 
54-090 

0.19976 
0.19972 
0.19968 
0.19963 
0.19959 

50-376 
Si-374 
52.371 

0.29923 
0.29907 
0.29892 
0.29877 
0.29862 

50.848 
51.842 

52-834 
53-826 
54.818 

0.39820 
0.39784 
0.39748 
0.39712 
0.39676 

51-506 
52-497 
53.486 

54-472 
55-458 

0.49656 

0.49585 
0.49514 

0-49444 
0-49374 

52-357 
53-342 

55-304 
56.281 

55 
56 
57 
58 
59 

0.099944 
0.099939 
0.099933 
0.099928 
0.099922 

55-089 
56.088 
57-086 
58.085 
59-083 

0-19955 
O.I995I 
0.19946 
O.I994I 
0.19937 

55-357 
56.352 
57-347 
58-342 
59-336 

0.29847 
0.29833 
0.29819 
0.29804 
0.29790 

55-809 
56.799 

58*773 
59.760 

0.39641 
0.39607 
0-39572 
0.39538 
0.39505 

56.440 
57-421 
58.400 
59-378 
60.354 

0.49305 
0.49238 
0.49172 
0.49106 
0.49041 

57-254 
58.226 

59-195 
60.160 
61.123 

6o 
61 
62 

63 

64 

0.099917 
0.099912 
0.099907 
0.099902 
0.099897 

60.082 
61.081 
62.079 
63-077 
64-075 

0.19934 
0.19929 
0.19925 
O.I992I 
0.19918 

60.331 
61.324 
62.317 
63.309 
64.301 

0.29777 
0.29764 
0.29751 
0.29738 
0.29725 

60.746 
61.731 
62.715 
63.698 
64.680 

0-39473 
0.39441 
0.39409 
0-39379 
0-39349 

61.330 
62.302 

63-273 
64.243 
65.212 

0.48978 
0.48916 
0.48855 
0.48795 
0.48738 

62.085 
63.042 
63.998 
64.950 
65.900 

65 
66 

67 
68 
69 

0.099893 
0.099889 
0.099885 
0.099881 
0.099877 

65-073 
66.071 
67.069 
68.066 
69.064 

0. 

o. 

0. 
0. 
0. 

19914 
19911 
19908 
19904 
19901 

65-293 
66.284 

67-275 
68.265 

69-255 

0.29712 
0.29700 
0.29689 
0.29678 
0.29667 

65.661 
66.641 
67.621 
68.599 
69-577 

0.39320 

o.39293 
0.39266 
0.39240 
0.39214 

66.179 
67.144 
68.108 
69.070 
70.030 

0.48681 
0.48627 

0.48575 
0.48523 
0.48473 

66.848 
67.794 
68.738 
69.679 
70.617 

70 

72 
73 

74 

0.099873 
0.099869 
0.099865 
0.099861 
0.099858 

70.062 
71-059 
72.057 
73-054 
74-051 

0.19898 

0.19895 

0.19892 
0.19889 
0.19887 

70.245 
7I-235 
72.225 

73-214 
74-203 

0.29657 
0.29647 
0.29637 
0.29628 
0.29620 

70-555 
71-532 
72.508 
73-483 

0.39190 
0.39167 

0.39*45 
0.39123 
0.39104 

70.990 
71.948 
72.905 
73-861 
74.817 

0.48426 
0.48380 
0.48338 
0.48296 
0.48257 

71-554 
72.489 
73.422 
74-354 
7S-284 

75 
76 
77 
78 
79 

0.099855 
0.099852 
0.099850 
0.099847 
0.099845 

75-048 
76.045 
77-042 
78.039 
79.036 

0.19885 
0.19883 
0.19881 
0.19878 
0.19876 

75-I92 
76.180 
77-168 
78-156 
79.144 

0.29612 
0.29604 
0.29596 
0.29590 
0.28584 

75-432 
76.406 

77-379 
78.351 
79-322 

0.39084 
0.39066 
0.39050 

0.39034 
0.39018 

75-771 
76.724 
77.676 
78.628 
79-578 

0.48220 
0.48184 
0.48152 
0.48121 
0.48093 

76.212 
77.138 
78.062 
78.986 
79.909 

80 
81 
82 

83 
84 

0.099843 
0.099841 
0.099839 
0.099838 
0.099837 

80.033 
81.030 
82.026 
83-023 
84.020 

0.19875 
0.19873 

0.19872 
0.19871 
0.19870 

80.131 
81.118 
82.106 

83-093 
84.080 

0.29578 

0.29573 
0.29569 
0.29566 
0.29563 

80.294 
81.266 
82.238 
83.209 
84.180 

0.39004 

0-38993 
0.38983 

0.38973 
0.38965 

80.528 

8i.477 
82.425 

83-373 
84.321 

0.48067 
0.48044 
0.48023 
0.48004 
0.47987 

80.830 
81.750 
82.669 
83-587 
84-505 

85 
86 

87 
88 

89 

0.099836 
0.099835 
0.099834 
0.099833 
0.099832 

85.017 
86.014 
87.011 
88.008 
89.004 

0.19869 

0.19868 
0.19868 
0.19867 
0.19867 

85-067 
86.054 
87.041 
88.028 
89.014 

0.29561 
0.29559 
0.29557 
0.29555 
0-29553 

85.150 
86.120 
87.090 
88.060 
89.030 

0.38958 
0.38952 
0.38948 
0.38946 
0.38944 

85.268 
86.215 
87.162 
88.108 
89.054 

0.47972 
0.47960 
0.47952 
0.47947 
0-47945 

85-422 
86.338 

87-254 
88.170 
89.085 

90 

0.099831 

90.000 

0.19867 

90.000 

0.29552 

90.000 

0.38942 

90.000 

0-47943 

90.000 

Note. 

sinh  (o  18)  =  o  /8. 

Examples. 

sinh  (0.3  765°)  =  o 

.29712  7j 

>5°.66i  = 

0.29712  / 

65°.39'.4 

o". 

sinh 


[2] 


TABLE  I.    HYPERBOLIC  SINES,     sinh 


=  r  /y.     CONTINUED 


0.6 


0.7 


0.8 


0.9 


i.o 


45 
46 
47 
48 
49 

0.60042  48.440 
0.59918  49-437 
0-59793  50.434 
0.59667  51.426 
0.59542  52.414 

0.70094  49.676 
0.69894  50.679 
0.69695  51.676 
0.69497  52.666 
0.69299  53.652 

0.80184  S^oS 
0.79885  52-112 
0.79587  53-109 
0.79291  54-099 
0.78996  55-o82 

0.90327  52.728 
0.89904  53.735 
0.89482  54.734 
0.89060  55.725 
0.88640  56.707 

1-00553  54-531 
0-99975  55-540 
0-99394  56.550 
0.98816  57-543 
0.98242  58.525 

50 
51 

52 

53 

54 

0.59418  53.398 
0.59295  54-379 
0.59174  55-355 
0.59053  56-326 
0.58932  57.293 

0.69102  54-632 
0.68907  55.606 
0.68713  56.574 
0.68521  57.537 
0.68331  58.493 

0.78703  56-058 
0.78412  57-026 
0.78124  57.987 
0.77838  58.940 
0-77555  59-886 

0.88224  57.679 
0.87810  58.642 
0.87400  59.595 
0.86993  60.538 
0.86590  61.472 

0.97672  59-495 
0.97105  60.453 
0.96543  61.399 
0.95986  62.333 
0-95435  63.255 

55 
56 
57 
58 
59 

0.58814  58.256 
0.58698  59.215 
0.58583  60.171 
0.58469  61.122 
0.58357  62.069 

0.68144  59-445 
0.67959  60.391 
0.67776  61.331 
0.67595  62.265 
0.67419  63.193 

0.77275  60.824 
0.76999  61.755 
0.76727  62.678 
0.76459  63.593 
0.76195  64.502 

0.86192  62.396 
0.85800  63.312 
0.85414  64.218 
0.85034  65.114 
0.84660.  66.000 

0.94890  64.166 
0-94353  65.065 
0.93825  65.952 
0.93305  66.827 
0.92795  67.691 

6o 
61 
62 
63 
64 

0.58249  63.013 
0.58142  63.953 
0.58037  64.889 
0.57934  65.821 
0.57834  66.749 

0.67247  64.117 
0.67078  65.036 
0.66912  65.951 
0.66749  66.859 
0.66591  67.762 

0.75938  65.405 
0.75686  66.300 
0-75439  67.189 
0.75197  68.070 
0.74962  68.944 

0.84295  66.878 
0.83937  67.747 
0.83587  68.607 
0.83244  69.458 
0.82909  70.300 

0.92295  68.544 
0.91805  69.385 

o-9I325  70-215 
0.90856  71.033 
0.90400  71.841 

65 
66 
67 
68 
69 

0-57737  67.674 
0.57643  68.596 
0-57553  69.515 
0.57465  70.430 
0-57379  7I-342 

0.66437  68.661 
0.66288  69.554 
0.66145  70.444 
0.66005  71.329 
0.65870  72.209 

0-74733  69.812 
0.74512  70.674 
0.74298  7i.529 
0.74091  72-379 
0.73891  73-223 

0.82585  71.136 
0.82270  71.962 
0.81967  72.780 
0.81672  73.590 
0.81387  74.392 

0.89957  72-637 
0.89527  73-424 
0.89111  74.201 
0.88708  74.968 
0.88320  75.723 

70 

72 
73 
74 

0.57297  72.251 
0.57219  73.157 
0.57H5  74-061 
0.57074  74-962 
0.57006  75.860 

0.65740  73-o85 
0.65616  73.957 
0.65498  74.825 

0-65385  75-689 
0.65278  76.550 

0.73698  74-061 
0.73513  74.894 
0-73337  75-722 
0.73169  76.544 
0.73009  77-361 

0.81114  75.187 
0.80853  75.975 
0.80602  76.756 
0.80363  77.530 
0.80137  78.298 

0.87947  76-469 
0.87589  77-207 
0.87247  77.936 
0.86921  78.656 
0.86612  79.368 

75 
76 
77 
78 
79 

0.56941  76-756 
0.56881  77.649 
0.56824  78.540 
0.56772  79.429 
0.56724  80.317 

0.65176  77.408 
0.65081  78.263 
0.64992  79.114 
0.64909  79.962 
0.64832  80.808 

0.72858  78.174 
0.72716  78.982 
0.72583  79-787 
0.72459  80.588 
0.72345  81.385 

0.79924  79.059 
0.79723  79-8i5 
0-79535  80.566 
0-79359  81.312 
0.79197  82.053 

0.86320  80.072 
0.86045  80.769 
0.85788  81.458 
0.85549  82.141 
0.85328  82.818 

80 
81 
82 
83 
84 

0.56679  81.203 
0.56638  82.087 
0.56602  82.970 
0.56570  83.852 
0.56542  84.732 

0.64761  81.652 
0.64697  82.493 
0.64640  83.332 
0.64589  84.169 
0.64545  85.005 

0.72241  82.179 
0.72146  82.969 
0.72061  83.757 
0.71985  84.543 
0.71919  85.327 

0.79048  82.789 
0.78913  83.522 
0.78792  84.251 
0.78685  84.976 
0.78592  85.699 

0.85125  83.489 
0.84940  84.156 
0.84774  84-817 
0.84628  85.474 
0.84501  86.128 

85 
86 

8? 
88 
89 

0.56518  85.612 
0.56498  86.490 
0.56483  87.368 
0.56473  88.246 
0.56466  89.123 

0.64507  85.839 
0.64476  86.673 
0.64452  87.506 
0.64435  88.338 
0.64425  89.169 

0.71863  86.109 
0.71817  86.889 
0.71781  87.668 
0.71755  88.446 
0.71740  89.223 

0.78513  86.420 
0.78448  87.138 
0.78397  87.855 
0.78361  88.571 
0.78339  89.286 

0.84393  86.779 
0.74305  87.426 
0.84236  88.071 
0.84186  88.715 
0.84156  89.358 

9P 

0.56464  90.000 

0.64422  90.000 

0.71736  90.000 

0.78333  90.000 

0.84147  OO.OPO 

Examples,     sinh  (i.o  /go")  =  0.84147  /go0. 

sinh"1  (0.87947  /76°.469)  =  1.0/70°. 


t3] 


TABLE  I.    HYPERBOLIC  SINES,     sinh  (p  IS)  =  r  {y.     CONTINUED 


1.4 


i-S 


45 

1.1089 

56.519 

1.2138 

58.692 

1.3205 

61.034 

1.4297 

63-568 

1-5418 

66.262 

46 

I.IOI2 

57-543 

1.2037 

59.726 

1.3078 

62.092 

1.4138 

64.639 

1.5222 

67-355 

47 

1-0935 

58.555 

I-I937 

60.748 

1.2951 

63.128 

1-3979 

65.689 

1.5027 

68.426 

48 

1.0858 

59-553 

1.1838 

6i.753 

1.2824 

64.142 

1.3822 

66.717 

1.4834 

69.474 

49 

1.0782 

60.536 

I-I739 

62.741 

1.2699 

65-I37 

1.3665 

67.723 

1.4642 

70.496 

50 

1.0706 

61.506 

1.1641 

63.712 

1.2574 

66.113 

1-3509 

68.707 

I-445I 

71.492 

5i 

1.0630 

62.461 

I-I543 

64.666 

1.2450 

67.068 

1-3355 

69.668 

1.4262 

72.462 

52 

1-0556 

63.401 

1.1446 

65.603 

1.2327 

68.004 

1.3202 

70.605 

1.4075 

73-405 

53 

1.0482 

64.327 

1-1350 

66.523 

1.2206 

68.919 

1-3051 

7i.5i9 

1.3889 

74-321 

54 

1.0409 

65-239 

1.1256 

67-425 

1.2086 

69.814 

1.2902 

72.409 

1.3706 

75.210 

55 

1.0336 

66.136 

1.1162 

68.310 

1.1967 

70.688 

1.2754 

73-275 

I.3525 

76.072 

56 

1.0265 

67.019 

1.1070 

69.178 

1.1850 

7I-542 

1.2609 

74.117 

1-3347 

76.907 

57 

1.0195 

67.888 

1.0979 

70.028 

I-I735 

72.376 

1.2466 

74-936 

1.3172 

77.713 

58 

1.0126 

68.742 

1.0890 

70.860 

1.1622 

73.188 

1.2325 

75-730 

1.3000 

78.491 

59 

1.0058 

69.581 

1.0802 

7I-675 

1.1511 

73.980 

1.2187 

76.500 

1.2831 

79.240 

6o 

0.99920 

70.406 

1.0716 

72.474 

1.1403 

74-752 

1.2052 

77.246 

1.2665 

79.964 

61 

0.99269 

71.218 

1.0632 

73-255 

1.1296 

75-502 

1.1919 

77.967 

1-2503 

80.657 

62 

0.98633 

72.016 

1.0550 

74-oi9 

1.1192 

76.232 

1.1790 

78.663 

1-2345 

81.321 

63 

0.98013 

72.800 

1.0470 

74.767 

1.1091 

76.942 

1.1664 

79-335 

1.2191 

81.956 

64 

0.97409 

73-570 

1.0392 

75-497 

1.0992 

77.632 

1.1541 

79-983 

1.2040 

82.562 

65 

0.96821 

74.3?7 

1.0316 

76.211 

1.0895 

78.301 

1.1421 

80.607 

1.1894 

83.140 

66 

0.96251 

75-07I 

1.0242 

76.909 

1.0802 

78.950 

1-1305 

81.207 

1.1752 

83.690 

67 

0.95698 

75-8oi 

1.0171 

77-590 

1.0712 

79.580 

1.1193 

81.783 

1.1615 

84.211 

68 

0.95165 

76-519 

I.OIO2 

78.257 

1.0625 

80.191 

1.1084 

82.335 

1.1482 

84.704 

69 

0.94650 

77.225 

1.0035 

78.907 

1.0540 

80.783 

1.0979 

82.865 

I.I354 

85.169 

70 

0.94156 

77-9i8 

0.99712 

79-543 

1.0459 

Si-357 

1.0879 

83.373 

1.1231 

85.607 

71 

0.93682 

78.600 

0.99099 

80.164 

1.0382 

81.912 

1.0783 

83.858 

1.1113 

86.018 

72 

0.93229 

79.271 

0.98514 

80.771 

1.0308 

82.450 

1.0691 

84.322 

1.  1000 

86.402 

73 

0.92798 

79-931 

0-97957 

81-365 

1.0237 

82.971 

1.0603 

84-765 

1.0893 

86.761 

74 

0.92388 

80.580 

0.97428 

81.946 

1.0170 

83.476 

1.0520 

85.188 

1.0791 

87.096 

75 

0.92001 

81.220 

0.96927 

82.514 

1.0107 

83.966 

1.0441 

85-59I 

1.0694 

87.406 

76 

0.91637 

81.850 

0.96456 

83.069 

1.0047 

84.440 

1.0367 

85-975 

1.0603 

87.693 

77 

0.91296 

82.471 

0.96015 

83.614 

0.99916 

84.900 

1.0298 

86.342 

1.0518 

87.958 

78 

0.90978 

83.084 

0.95605 

84.148 

0.99397 

85-346 

1.0233 

86.692 

1.0439 

88.201 

79 

0.90685 

83.689 

0.95226 

84.672 

0.98917 

85-780 

1.0173 

87.026 

1.0366 

88.425 

80 

0.90416 

84.286 

0.94878 

85-187 

0.98477 

86.202 

1.0119 

87.346 

1.0299 

88.630 

81 

0.90172 

84.877 

0.94562 

85-693 

0.98077 

86.614 

1.0069 

87.651 

1.0238 

88.818 

82 

0-89953 

85.462 

0.94279 

86.191 

0.97715 

87.016 

1.0024 

87.944  . 

1.0183 

88.991 

83 

0.89759 

86.041 

0.94029 

86.683 

0.97400 

87.408 

0.99845 

88.227 

1.0135 

89.149 

84 

0.89590 

86.616 

0.93811 

87.169 

0.97124 

87-793 

0.99502 

88.499 

1.0093 

89.294 

85 

0.89447 

87.186 

0.93626 

87.649 

0.96890 

88.171 

0.99210 

88.762 

1.0057 

89.429 

86 

0.89330 

87-753 

0-93474 

88.124 

0.96698 

88.544 

0.98971 

89.018 

1.0027 

89-554 

87 

0.89238 

88.317 

0.93356 

88.596 

0.96548 

88.912 

0.98785 

89.269 

1.0004 

89-673 

88 

0.89173 

88.879 

0.93272 

89.065 

0.96441 

89.277 

0.98652 

89-515 

0.99880 

89-785 

89 

0.89134 

89.440 

0.93221 

89-533 

0.96377 

89.639 

0.98572 

89.758 

0.99782 

89.893 

DO 

0.89121 

90.000 

0.93204 

90.000 

0.96356 

90.000 

0.98545 

90.000 

0.99749 

90.000 

Examples. 

sinh  (i. 

i/45°)  = 

1.1089  /j 

;6°.si9  =  1.1089  /s6°.3i'.o8'. 

sinh"1  (1.1084  /82°.335)  = 

1.4  /68°. 

[4] 


TABLE  I.    HYPERBOLIC  SINES,     sinh  (p  /5)  =  r  /%    CONTINUED 


1.6 


1.8 


1.9 


45 

I-6575 

69.117 

1.7776 

72.133 

1.9029 

75.292 

2-0343 

78.590 

2.1726 

82.016 

46 

1-6338 

70.241 

1-7493 

73-288 

1.8693 

76.486 

1.9947 

79.829 

2.1266 

83-304 

47 

1.6103 

71-339 

1.7210 

74.418 

I-8359 

77.651 

1-9554 

81.037 

2.0809 

84.560 

48 

1.5868 

72.409 

1.6929 

75-5I5 

1.8027 

78.784 

1.9165 

82.2IO 

2.0356 

85.778 

49 

I-563S 

73-451 

1.6651 

76.582 

1.7697 

79.883 

1.8779 

83.346 

1.9907 

86.958 

5° 

1.5404 

74-465 

I-6375 

77.618 

1-7370 

80.946 

1.8396 

84.444 

1.9462 

88.098 

51 

I-5I75 

75-449 

1.6102 

78.620 

1.7046 

81.974 

1.8016 

85.503 

1.9022 

89.196 

52 

1.4949 

76.402 

1-5831 

79-590 

1.6726 

82.966 

1-7643 

86.524 

1.8588 

90-253 

53 

I-4725 

77-325 

1-5563 

80.527 

1.6410 

83.921 

1-7273 

87.505 

1.8160 

91.267 

54 

1.4504 

78.218 

1-5299 

81.429 

1.6098 

84.839 

1.6909 

88.444 

1-7737 

92.236 

55 

1.4285 

79.079 

1-5039 

82.296 

I-579I 

85-718 

1.6550 

89.342 

1.7320 

93.160 

56 

1.4070 

79.910 

1.4782 

83.128 

1.5488 

86.558 

1.6196 

90.197 

1.6911 

94.039 

57 

1-3859 

80.709 

1-4530 

83-924 

1.5190 

87.358 

1.5848 

9I.OO8 

1.6509 

94.869 

58 

1-3651 

Si-475 

1.4282 

84.683 

1.4898 

88.116 

i.S5o6 

91-773 

1.6114 

95-650 

59 

1-3447 

82.209 

1.4039 

85.406 

1.4611 

88.834 

1-5171 

92.493 

1.5726 

96.382 

60 

I-3247 

82.910 

1.3800 

86.091 

1-4330 

89.510 

1.4843 

93.167 

1-5347 

97.062 

61 

1-3051 

83.578 

1-3567 

86.738 

1-4055 

90.142 

1-4523 

93.792 

1.4976 

97-688 

62 

1.2860 

84.212 

1-3339 

87-347 

1.3787 

90.730 

1.4210 

94.368 

1.4614 

98.261 

63 

1.2674 

84.813 

1.3117 

87.917 

I.3525 

91.274 

•3904 

94.893 

1.4260 

98.778 

64 

1.2492 

85-380 

1.2901 

88.447 

1.3270 

Qi-774 

.3606 

95.368 

i-39J5 

99.238 

65 

1.2316 

85-9I3 

1.2690 

88.938 

1.3022 

92.228 

•33i6 

95-792 

i.358o 

99.639 

66 

1.2145 

86.413 

1.2486 

89.390 

1.2781 

92.636 

•3035 

96.162 

I.325S 

99.980 

t57 

1.1979 

86.879 

1.2288 

89.802 

1.2548 

92.997 

.2762 

96.478 

1.2940 

100.261 

68 

1.1819 

87.311 

1.2097 

90.175 

1.2322 

93-3*2 

.2499 

96.741 

1.2635 

100.479 

69 

1.1664 

87.710 

1.1913 

90.508 

1.2104 

93-58o 

.2244 

96.949 

1.2340 

100.634 

70 

1.1516 

88.076 

1.1736 

90.8OI 

1.1895 

93.802 

.1999 

97.102 

1.2056 

100.725 

7i 

I-I373 

88.410 

1.1566 

9  LOSS 

1.1694 

93.976 

.1764 

97.199 

1.1783 

100.750 

72 

1.1237 

88.712 

1.1403 

91.271 

1.1502 

94.104 

•1539 

97.241 

1.1521 

100.709 

73 

1.1107 

88.982 

1.1248 

91.448 

1.1318 

94.187 

•  1324 

97.227 

1.1271 

100.602 

74 

1.0984 

89.221 

I.  HOI 

91.588 

1.1144 

94-224 

1.1119 

97.I58 

1.1033 

100.428 

75 

1.0867 

89.431 

1.0961 

91.692 

1.0979 

94-215 

1.0925 

97-034 

1.0807 

100.187 

76 

1-0757 

89.613 

1.0829 

91.760 

1.0823 

94.162 

1-0743 

96.856 

1.0594 

99.880 

77 

1.0654 

89.767 

1.0706 

91.794 

1.0677 

94.068 

1.0571 

96.625 

1-0394 

99.507 

78 

1.0558 

89.894 

I.059I 

9!-795 

1.0541 

93-933 

1.0411 

96.343 

1.0207 

99.070 

79 

i  .0469 

89.997 

1.0485 

9I-765 

1-0415  . 

93-7S8 

1.0262 

96.OI2 

1.0033 

98.571 

80 

1.0388 

90.076 

1.0388 

91-705 

1.0299 

93-545 

1.0126 

95-633 

0.98729 

98.011 

81 

1.0314 

90.134 

1.0299 

91.618 

1.0194 

93.298 

I.OOO2 

95.209 

0.97271 

97-394 

82 

1.0248 

90.171 

I.O2I9 

9I-505 

1.0099 

93.019 

0.98899 

94-744 

0.95958 

96.722 

83 

1.0189 

90.191 

I.OI49 

91.370 

1.0015 

92.711 

0.97907 

94.240 

0.94792 

96.000 

84 

1.0138 

90.194 

1.0088 

91.214 

0.99422 

92-374 

0.97043 

93-702 

0-93775 

95-233 

85 

1.0095 

90.184 

I.OO36 

91.039 

0.98802 

92.015 

0.96309 

93-133 

0.92911 

94.426 

86 

i.  0060 

90.161 

0.99929 

90.850 

0.98293 

91.636 

0.95706 

92-539 

0.92201 

93-584 

87 

1.0032 

90.129 

0.99596 

90.649 

0.97896 

91.241 

0.95236 

91.923 

0.91646 

92.713 

88 

I.OOI2 

90.090 

0-99357 

90.438 

0.97612 

90.834 

0.94899 

91.291 

0.91248 

91.821 

89 

I.OOOO 

90.046 

0.99214 

90.220 

0.97442 

90.419 

0.94697 

90.648 

0.91009 

90.914 

90 

Q-99957 

90.000 

0.99166 

90.000 

0.97385 

90.000 

0.94630 

90.000 

0.90930 

90.000 

Examples,     sinh  (1.6  /6o°)  =  1.3247  /82°.9io  =  1.3247  /82°.S4'.36V 
sinh"1  (1.1999  /97°.iQ2)  =  1.9  /7Q°. 


[Si 


TABLE  I.    HYPERBOLIC  SINES,     sinh  (p  /5)  =  r  /jy.    CONTINUED 


2-3 


2.4 


2-5 


45 

2.3190 

85.558 

2-4745 

89.205 

2.6404 

92.946 

2.8177 

96.769 

3.0079 

100.661 

46 

2.2658 

86.905 

2.4135 

90.613 

2.5707 

94.419 

2.7386 

98.312 

2.9185 

102.278 

47 

2.2131 

88.213 

2-3530 

91.981 

2.5017 

95-85i 

2.6603 

99.813 

2.8301 

103.852 

48 

2.1608 

89.482 

2.2936 

93-307 

2-4334 

97.241 

2.5829 

101.271 

2.7429 

105-383 

49 

2.1090 

90.711 

2.2337 

94.592 

2.3659 

98.588 

2.5065 

102.685 

2.6568 

106.869 

SO 

2.0577 

91.898 

2.1750 

95.834 

2.2992 

99.890 

2.4311 

104.053 

2.5720 

108.309 

Si 

2.0071 

93.042 

2.1171 

97.031 

2.2334 

101.146 

2.3568 

105.375 

2.4885 

109.701 

52 

I-957I 

94.142 

2.0600 

98.181 

2.1685 

102.354 

2.2836 

106.648 

2.4064 

111.044 

S3 

1.9078 

95-I97 

2.0037 

99.284 

2.1046 

103.514 

2.2117 

107.871 

2-3257 

1I2-337 

54 

1.8592 

96.205 

1.9483 

100.338 

2.0418 

104.623 

2.1410 

109.042 

2.2465 

"3-578 

55 

1.8114 

97-i65 

1.8938 

101.342 

1.9801 

105.680 

2.0714 

110.160 

2.1687 

114.765 

56 

1.7644 

98.076 

1.8402 

102.294 

I-9J95 

106.683 

2.0032 

111.223 

2.0925 

115.896 

57 

1.7182 

98.935 

1.7876 

103.193 

1.  8600 

107.629 

1.9364 

112.228 

2.0178 

116.970 

58 

1.6729 

99.742 

1.7360 

104.035 

1.8016 

108.518 

1.8709 

"3-174 

1.9447 

117-984 

59 

1.6284 

100.494 

1.6854 

104.820 

1-7445 

109.347 

i.  8068 

114.059 

1-8733 

118.936 

60 

1.5849 

101.191 

I-6359 

105.546 

1.6886 

110.114 

1.7441 

114.880 

1.8035 

119.823 

61 

I-5424 

101.830 

I.5875 

106.210 

1.6340 

110.816 

1.6829 

115-634 

1-7353 

120.642 

62 

1.5008 

102.410 

1.5402 

106.811 

1.5807 

111.451 

1.6232 

116.319 

1.6688 

121.391 

63 

1.4603 

102.929 

1.4941 

107.345 

1.5287 

112.016 

1.5649 

116.931 

1.6040 

122.067 

64 

1.4208 

103.386 

1.4492 

107.811 

1.4780 

112.509 

1.5081 

117.467 

1.5409 

122.665 

65 

1.3824 

103.777 

1-4055 

108.207 

1.4286 

112.926 

1.4528 

117.924 

1.4794 

123.182 

66 

J-345i  . 

104.101 

1.3630 

108.529 

1.3806 

113.264 

I-399I 

118.297 

1.4196 

123.613 

67 

1.3089 

104.357 

1.3218 

108.775 

I-334I 

"3-519 

1.3469 

118.583 

1.3616 

123-954 

68 

1.2738 

104.542 

1.2819 

108.943 

1.2890 

113.688 

1.2963 

118.777 

1-3052 

124.198 

69 

1.2399 

104.655 

1-2433 

109.029 

1-2453 

113.767 

1-2473 

118.874 

1.2506 

124.341 

70 

1.2072 

104.694 

1.2060 

109.030 

1.2031 

"3-752 

1.1999 

118.868 

1.1977 

124.376 

7i 

1.1758 

104.656 

1.1701 

108.944 

1.1624 

113.638 

1.1541 

118.754 

1.1466 

124.296 

72 

I-I457 

104.541 

I-I356 

108.769 

1.1232 

113.422 

1.1099 

118.526 

1.0972 

124.092 

73 

1.1168 

104.348 

1.1026 

108.501 

1.0857 

113.100 

1.0675 

118.177 

1.0497 

123-755 

74 

1.0893 

104.074 

1.0711 

108.138 

1.0498 

112.667 

1.0268 

117.701 

1.0040 

123.277 

75 

1.0632 

103.719 

1.0411 

107.678 

i-oi55 

112.118 

0.98795 

117.091 

0.96019 

122.648 

76 

1-0385 

103.283 

1.0126 

107.119 

0.98292 

111.449 

0.95090 

116.341 

0.91831 

121.857 

77 

1-0153 

102.766 

0.98581 

106.459 

0.95214 

110.659 

0.91576 

II5-443 

0.87843 

120.891 

78 

0-99353 

102.168 

0.96065 

105.699 

0.92320 

109.744 

0.88261 

114.390 

0.84063 

119.740 

79 

0.97332 

101.491 

0.93722 

104.839 

0.89615 

108.701 

0.85152 

113.178 

0.80503 

118.392 

80 

0.95468 

100.736 

0.91557 

103.879 

0.87109 

107-531 

0.82256 

111.803 

0.77173 

116.836 

81 

0-93765 

99907 

0-89575 

102.822 

0.84808 

106.233 

0.79589 

110.261 

0.74083 

115.063 

82 

0.92229 

99.008 

o  87784 

101.671 

0.82719 

104.810 

0.77158 

108.550 

0.71251 

113.065 

83 

0.90862 

98.043 

0.86187 

100.432 

0.80853 

103.267 

0.74974 

106.676 

0.68692 

110.841 

84 

0.89668 

97-oiS 

0.84789 

99.111 

0.79215 

101.611 

0-73051 

104.643 

0.66422 

108.390 

85 

0.88652 

95-933 

0.83596 

97-7I5 

0.77813 

99.851 

0.71396 

102.462 

0.64461 

105.723 

86 

0.87817 

94.806 

0.82613 

96.254 

0.76655 

98.000 

0.70026 

100.147 

0.62827 

102.855 

87 

0.87164 

93-639 

0.81845 

94.740 

0-75747 

96.071 

0.68948 

97.717 

0.61536 

99.811 

88 

0.86696 

92.443 

0.81293 

93-183 

0.75095 

94.082 

0.68171 

95-197 

0.60602 

96.624 

89 

0.86414 

91.226 

0.80961 

91-599 

0.74702 

92.051 

0.67703 

92.615 

0.60036 

93-338 

90 

0.86321 

90.000 

0.80850 

90.000 

0-74571 

90.000 

0.67546 

90.000 

0.59847 

90.000 

Examples,     sinh  (2.5  775°  =  0.96019  /i22°.648  =  0.96019  /i220.38'.53g. 
sinh"1  (0.60036  /93°-338)  =  2.5  789°. 


[6J 


TABLE  I. 

2.6 

HYPERBOLIC  SINES,  sinh  (p  IS)  =  r  /T.  CONTINUED 
2.7            2.8            2.9           3.0 

o 

o 

o 

o 

0 

o 

4S 

3.2121 

104.613 

3-43i8 

108.614 

3-6685 

112.653 

3-9236 

116.721 

4.1986 

120.814 

46 

3-HI5 

106.307 

3-3*91 

110.386 

3-5426 

114.506 

3-7832 

118.658 

4.0426 

122.832 

47 

3.0123 

107.957 

3.2079 

112.  Il6 

3.4186 

116.317 

3-6453 

120.551 

3-8895 

124.808 

48 

2.9144 

109.564 

3-0985 

II3.80I 

3.2966 

118.084 

3-5098 

122.400 

3-7394 

126.741 

49 

2.8179 

111.126 

2.9908 

115.442 

3.1767 

119.806 

3.3768 

124.205 

3-5923 

128.630 

5° 

2.7229 

112.641 

2.8849 

117.037 

3.0590 

121.483 

3-2465 

125.965 

3-4483 

130.474 

5i 

2.6295 

114.109 

2.7809 

118.585 

2.9436 

123.113 

3.1189 

127.679 

3.3076 

132.272 

S2 

2.5378 

115.528 

2.6789 

120.084 

2.8306 

124.694 

2.9941 

129.345 

3.1701 

134.024 

53 

2.4478 

116.897 

2-5789 

121.532 

2.720O 

126.226 

2.8721 

130.963 

3-0359 

135.729 

54 

2-3595 

118.214 

2.4809 

122.929 

2.6118 

127.708 

2.7529 

132.532 

2.9051 

137.386 

55 

2.2729 

119.476 

2.3850 

124.274 

2.5060 

129.138 

2.6366 

134-050 

2.7777 

138.994 

56 

2.1882 

120.684 

2.2913 

125.563 

2.4027 

130.515 

2.5232 

135-517 

2.6536 

140.553 

57 

2-1053 

121.834 

2.1996 

126.797 

2.3019 

131-837 

2.4127 

136.932 

2-5330 

142.062 

58 

2.0242 

122.925 

2.IIOI 

127.973 

2.2036 

133-103 

2-3051 

138.293 

2.4157 

143-519 

59 

1.9450 

123.954 

2.0228 

129.088 

2.1077 

134-3" 

2.2004 

139.598 

2.3018 

144.924 

60 

1.8677 

124.918 

1-9377 

130.140 

2.0144 

135-459 

2.0986 

140.847 

2.1912 

146.276 

61 

1-7923 

125.816 

I.8547 

I3I.I28 

1.9236 

136.544 

1.9997 

142.037 

2.0839 

147-574 

62 

1.7187 

126.644 

1-7739 

132.047 

1-8352 

137-565 

1.9036 

143.167 

1.9798 

148.817 

63 

1.6470 

127.399 

1-6952 

132.895 

1-7493 

138.519 

1.8103 

144.235 

1.8789 

150.004 

64 

1-5772 

128.077 

1.6187 

133.669 

1.6658 

I39-403 

1.7198 

145-239 

1.7812 

151-133 

65 

1-5094 

128.674 

1-5442 

I34-364 

I-5847 

140.214 

1.6319 

146.176 

1.6865 

152.204 

66 

1-4435 

129.185 

1.4719 

134.978 

1.5060 

140.948 

1-5467 

147.044 

1.5948 

I53-2I4 

67 

1-3794 

129.606 

1.4016 

I35-504 

I-4295 

141.599 

1.4641 

147.838 

1.5061 

154.161 

68 

1.3171 

129.930 

J-3334 

135-935 

1-3553 

142.164 

1.3840 

148.555 

1.4202 

I55-043 

69 

1.2568 

130.151 

1.2673 

136.266 

1-2833 

142.634 

1.3063 

149.190 

1-3370 

155-858 

70 

1-1983 

130.262 

1.2031 

136.489 

1.2136 

I43-005 

1.2310 

149.738 

1.2565 

156.602 

7i 

1.1417 

130.254 

1.1409 

136.596 

1.1460 

143.267 

1.1581 

150.191 

1.1785 

157.271 

72 

1.0870 

130.117 

1.0807 

136.576 

1.0804 

143.411 

1.0874 

150.541 

1.1030 

157.860 

73 

1.0341 

129.841 

1.0226 

136.415 

1.0169 

143.424 

1.0189 

150.780 

1.0298 

158.363 

74 

0.98316 

129.414 

0.96631 

I36.IOI 

0.95550 

143.292 

0.95251 

150.894 

0.95893 

158.772 

75 

0.93420 

128.822 

0.91207 

I35-6I7 

0.89606 

142.996 

0.88819 

150.867 

0.89022 

159.077 

76 

0.88720 

128.049 

0.85985 

134-943 

0.83862 

142.516 

0.82587 

150.680 

0.82356 

159-265 

77 

0.84227 

127.079 

0.80965 

134.058 

0.78319 

141.827 

0.76551 

150.309 

0.75887 

I59-3I9 

78 

0.79948 

125.893 

0.76157 

I32-935 

0.72979 

140.897 

0.70706 

149.721 

0.69605 

159.216 

79 

0.75891 

124.471 

0.71568 

I3I-S42 

0.67847 

139.686 

0.65054 

148.876 

0.63503 

158.925 

80 

0.72068 

122.793 

0.67215 

129.847 

0.62934 

138.145 

0-59597 

147.721 

0-57573 

158.404 

81 

0.68499 

120.837 

0.63109 

I27.8l2 

0.58252 

136.217 

0.54340 

146.183 

0.51809 

I57.592 

82 

0.65203 

118.583 

0.59275 

125.396 

0.53822 

I33-833 

0.49294 

144.169 

0.46213 

156.404 

83 

0.62196 

116.016 

0-55739 

122.554 

0.49674 

130.908 

0.44484 

141-553 

0.40789 

154.709 

84 

0.59507 

113.126 

0.52536 

119.252 

0.45849 

127-351 

0.39946 

138.168 

0.35548 

152-313 

85 

0.57164 

109.910 

0.49705 

II5-458 

0.42399 

123.066 

0.35738 

133-795 

0.30521 

148-903 

86 

0.55191 

106.382 

0.47291 

III.l64 

0-39394 

117.968 

0.31946 

128.168 

0.25775 

143-974 

87 

0.53621 

102.571 

0-45344 

106.390 

0.36915 

112.015 

0.28691 

121.005 

0.21433 

136.701 

88 

0.52479 

98.524 

0.43912 

IOI.I94 

0.35055 

105.244 

0.26143 

112.  IIO 

0.17731 

125.843 

89 

0.51783 

94-307 

0.43033 

95-682 

0-33893 

97.809 

0.24496 

101.576 

0.15092 

110.091 

90      0.51550     90.000      0.42738     90.000      0.33499      90.000      0.23925      90.000      0.14112      90.000 


Examples,     sinh  (3.0  /5o°)  =  3.4483  / 
sinh""1  (0.15092  /iio°.Q9i) 


SQ°-474  =  3-4483  /i3Q°-28/.26>. 
-  3-Q  /89°- 


[7] 


TABLE  II.    HYPERBOLIC   COSINES,     cosh  (p  /5)  =  r  /_y 


0-3 


0.4 


45 

I.OOOOI 

0.287 

I.OOOI3 

1.148 

1.00067 

2.578 

I.O02IO 

4.578 

1.00519 

7.141 

46 

0.99983 

0.287 

0-99943 

1.146 

0.99910 

2-577 

0-99933 

4-584 

1.00085 

7-159 

47 

0.99966 

0.286 

0.99873 

1.144 

0-99753 

2.576 

0.99655 

4-584 

0.99649 

7.166 

48 

0.99948 

0.285 

0.99804 

1.141 

0-99597 

2-571 

0.99378 

4.578 

0.99214 

7-165 

49 

0.99931 

0.283 

0-99735 

1.136 

0.99441 

2.562 

0.99101 

4.566 

0.98780 

7-155 

5° 

0.99914 

0.282 

0.99666 

1.131 

0.99285 

2-551 

0.98824 

4-551 

0.98347 

7-137 

51 

0.99897 

0.280 

0-99597 

1.123 

0.99131 

2-536 

0.98547 

4-528 

0.97917 

7.109 

52 

0.99880 

0.278 

0.99529 

1-115 

0.98977 

2-519 

0.98274 

4.500 

0.97490 

7-073 

S3 

0.99863 

0.275 

0.99462 

1.105 

0.98825 

2.498 

0.98003 

4.467 

0.97065 

7.028 

54 

0.99846 

0.272 

0-99395 

1.094 

0.98674 

2-474 

0-97734 

4427 

0.96644 

6-973 

55 

0.99830 

0.269 

0.99329 

1.081 

0.98525 

2.447 

0.97468 

4-382 

0.96226 

6.910 

56 

0.99814 

0.265 

0.99263 

1.067 

0.98377 

2.417 

0.97205 

4-332 

0.95814 

6.838 

57 

0.99798 

0.261 

0.99199 

1.052 

0.98232 

2-383 

0.96945 

4.276 

0.95406 

6.756 

58 

0.99782 

0-257 

0.99135 

1.036 

0.98089 

2-347 

0.96690 

4.214 

0.95005 

6.666 

59 

0.99766 

0-253 

0.99073 

1.018 

0.97948 

2.308 

0.96438 

4.147 

0.94609 

6.567 

60 

0.99751 

0.249 

0.99012 

0.999 

0.97810 

2.267 

0.96191 

4-075 

0.94219 

6.460 

61 

0.99736 

0.244 

0.98952 

0.979 

0.97674 

2.221 

0.95948 

3-998 

0.93838 

6-343 

62 

0.99721 

0.238 

0.98893 

0-957 

0.97541 

2-173 

0.95711 

3-9I4 

0-93465 

6.217 

63 

0.99707 

0.232 

0.98835 

0-934 

0.97411 

2.123 

0.85478 

3-826 

0.93099 

6.083 

64 

0.99693 

0.226 

0.98780 

0.910 

0.97285 

2.07O 

0-95251 

3-733 

0.92741 

5-941 

65 

0.99679 

O.22O 

0.98725 

0.885 

0.97163 

2.014 

0.95031 

3-635 

0.92393 

5-789 

66 

0.99666 

O.2I4 

0.98672 

0.859 

0.97044 

1-955 

0.94816 

3-532 

0-92054 

5-631 

67 

0.99653 

0.207 

0.98621 

0.832 

0.96927 

1.894 

0.94608 

3-423 

0.91725 

5-463 

68 

0.99641 

O.2OO 

0.98571 

0.804 

0.96814 

1.830 

0.94407 

3-3io 

0.91407 

5.288 

69 

0.99629 

0.193 

0.98523 

0-775 

0.96706 

1.764 

0.94213 

3-!93 

0.91100 

5.106 

70 

0.99617 

0.185 

0.98477 

0.744 

0.96601 

1.696 

0.94026 

3.072 

0.90805 

4.916 

7i 

0.99606 

0.177 

0.98433 

0.713 

0.96500 

1.626 

0.93846 

2.946 

0.90521 

4.718 

72 

0.99596 

0.169 

0.98391 

0.681 

0.96404 

1-553 

0.93674 

2.816 

0.90248 

4-5I3 

73 

0.99586 

0.161 

0.98350 

0.648 

0.96313 

1.479 

0.93510 

2.682 

0.89989 

4.302 

74 

0.99576 

0.153 

0.98312 

0.614 

0.96227 

1.402 

0-93354 

2-544 

0.89742 

4.085 

75 

0.99567 

0.144 

0.98276 

0.580 

0.96145 

1-324 

0.93207 

2.403 

0.89508 

3-86i 

76 

0.99559 

0-135 

0.98242 

0-545 

0.96068 

1.244 

0.93069 

2.258 

0.89288 

3-631 

77 

0-99551 

0.126 

0.98210 

0.509 

0.95996 

1.162 

0.92938 

2.  Ill 

0.89081 

3-396 

78 

0-99544 

0.117 

0.98181 

0.472 

0.95929 

1.078 

0.92816 

1.960 

0.88889 

3-155 

79 

0-99537 

0.108 

0.98154 

0-435 

0.95866 

0-993 

0.92705 

1.807 

0.88711 

2.910 

80 

0.99531 

0.099 

0.98128 

0-397 

0.95808 

0.908 

0.92603 

1.652 

0.88548 

2.660 

81 

0.99525 

0.090 

0.98105 

0-359 

0.95756 

0.821 

0.92509 

1.494 

0.88399 

2.406 

82 

0.99520 

0.080 

0.98085 

0.320 

0.95710 

0.732 

0.92425 

1-333 

0.88266 

2.149 

83 

0-995  1  5 

0.070 

0.98067 

0.281 

0.95670 

0.643 

0.62351 

1.170 

0.88147 

1.888 

84 

0.99511 

0.060 

0.98051 

0.242 

0-95635 

0-552 

0.92287 

i.  006 

0.88044 

1.624 

85 

0.99508 

0.050 

0.98037 

0.203 

0.95605 

0.461 

0.92233 

0.841 

0.87957 

1-357 

86 

0.99506 

0.040 

0.98026 

0.163 

0.95580 

0.369 

0.92189 

0.675 

0.87886 

1.087 

87 

0.99504 

0.030 

0.98018 

0.123 

0.95560 

0.277 

0.92154 

0.508 

0.87830 

0.816 

88 

0.99502 

0.02O 

0.98012 

0.082 

0-95545 

0.185 

0.92128 

0.340 

0.87790 

0-544 

89 

0.99501 

O.OIO 

0.98009 

0.041 

0-95537 

0.093 

0.92112 

0.171 

0.87766 

0.272 

90      0.99500   o.ooo      0.98007   o.ooo      0.95534   o.ooo      0.92106   o.ooo      0.87758   o.ooo 

Note.  cosh  (o/5)  =  i.o  /o^for  all  values  of  d. 

Examples,     cosh  (0.5  /8i°)  =  0.88399  /2°.4o6  =  0.88399  /2°.24/.22r. 
cosh"1  (0.97810  /2°.267)  =  0.3  /6o°. 

[8] 


TABLE  II.    HYPERBOLIC   COSINES,     cosh  (p  /8)  =  r  /T.     CONTINUED 


0.6 


0.7 


0.8 


0.9 


45 

1.01070 

10.254 

1.01982 

13.890 

1.03360 

18.010 

1.05333 

22.567 

1.08031 

27.487 

46 

1.00449 

10.291 

1.01136 

13.960 

1.02263 

18.132 

1.03959 

22.755 

1.06358 

27.762 

47 

0.99825 

10.315 

1.00289 

14.013 

1.01164 

18.231 

1.02583 

22.919 

1  .04680 

28.011 

48 

0.99199 

10.327 

0.99441 

14.050 

1.00063 

18.309 

1.01203 

23-059 

1.03000 

28.235 

49 

0-98575 

10.326 

0.98594 

14.071 

0.98963 

18.368 

0.99822 

23.176 

1.01315 

28.433 

50 

0-97953 

10.313 

0.97748 

14-075 

0.97864 

18.405 

0.98443 

23.267 

0.99632 

28.603 

Si 

0-97333 

10.287 

0.96906 

14.061 

0.96768 

18.421 

0.97064 

23-332 

0.97950 

28.743 

52 

0.96716 

10.248 

0.96066 

14.031 

0.95675 

18.414 

0.95690 

23-372 

0.96270 

28.854 

53 

0.96103 

10.196 

0.95232 

13.982 

0.94587 

18.384 

0.94320 

23-383 

0.94596 

28.933 

54 

0-95495 

10.131 

0.94404  / 

13.916 

0.93506 

18.332 

0.92957 

23-367 

0.92928 

28.981 

55 

0.94893 

10.053 

0.93582 

13-831 

0.92432 

18.256 

0.91603 

23.321 

0.91267 

28.994 

56 

0.94296 

9.961 

0.92768 

13.728 

0.91367 

18.156 

0.90257 

23.246 

0.89615 

28.974 

57 

0.93706 

9-856 

0.91962 

13.606 

0.90312 

18.031 

0.88922 

23.140 

0.87976 

28.917 

58 

0.93125 

9.738 

0.91166 

13-465 

0.89270 

17.881 

0.87602 

23.003 

0.86350 

28.823 

59 

0.92552 

9.606 

0.90382 

13.306 

0.88240 

17-705 

0.86294 

22.833 

0.84739 

28.689 

6o 

0.91987 

9.461 

0.89607 

13.127 

0.87222 

17.505 

0.85001 

22.631 

0.83142 

28.518 

61 

0.91433 

9-303 

0.88846 

12.929 

0.86221 

17.278 

0.83727 

22.394 

0.81564 

28.303 

62 

0.90889 

9-J3i 

0.88100 

12.713 

0.85237 

17.024 

0.82471 

22.122 

0.80008 

28.047 

63 

0.90357 

8-945 

0.87369 

12.476 

0.84271 

16.743 

0.81237 

2I.8I4 

0.78475 

27.745 

64 

0.89838 

8.747 

0.86653 

12.  22O 

0.83324 

16.435 

0.80025 

21.471 

0.76966 

27.396 

65 

0.89332 

8.536 

0.85954 

"•945 

0.82398 

16.100 

0.78837 

21.091 

0.75484 

26.999 

66 

0.88838 

8.312 

0.85272 

11.652 

0.81494 

15-738 

0.77675 

20.673 

0.74029 

26.555 

67 

0.88358 

8.076 

0.84609 

n-339 

0.80613 

15-349 

0.76540 

2O.2I8 

0.72603 

26.061 

68 

0.87894 

7.827 

0.83966 

11.007 

0.79757 

I4-93I 

0-75435 

19.722 

0.71212 

25-5I3 

69 

0.87445 

7.565 

0.83344 

10.656 

0.78927 

14.486 

0.74362 

I9.I88 

0.69857 

24.911 

70 

0.87012 

7.292 

0.82744 

10.287 

0.78125 

14.014 

0.73322 

I8.6I5 

0.68539 

24-254 

71 

0.86596 

7.007 

0.82166 

9.900 

0.77352 

I3-5I5 

0.72317 

iS.OOI 

0.67261 

23-54I 

72 

0.86197 

6.711 

0.81611 

9.496 

0.76610 

12.990 

0.71347 

I7.348 

0.66026 

22.772 

73 

0.85816 

6.404 

0.81081 

9-075 

0.75899 

12.438 

0.70416 

16.656 

0.64836 

21.946 

74 

0.85454 

6.086 

0.80576 

8.637 

0.75220 

11.860 

0.69527 

15.926 

0.63692 

21.061 

75 

0.85  in 

5.758 

0.80097 

8.183 

0.74575 

11.257 

0.68681 

I5-I56 

0.62599 

20.115 

76 

0.84787 

5.420 

0.79646 

7.7I3 

0.73965 

10.630 

0.67878 

14-347 

0.61560 

19.111 

77 

0.84484 

5-073 

0.79222 

7.228 

0.73392 

9-979 

0.67121 

I3.502 

0.60577 

18.049 

78 

0.84201 

4.718 

0.78826 

6.729 

0.72856 

9.306 

0.66412 

12.621 

0.59652 

16.929 

79 

0.83939 

4-355 

0.78459 

6.217 

0.72359 

8.612 

0.65753 

II.7O6 

0.58790 

15-753 

80 

0.83698 

3-984 

0.78121 

5-694 

0.71901 

7.897 

0.65145 

10-755 

0.57991 

14.521 

81 

0.83479 

3.606 

0.77814 

5-!59 

0.71484 

7.164 

0.64589 

9.776 

0.57259 

I3-237 

82 

0.83282 

3.221 

0.77538 

4-613 

0.71108 

6.413 

0.64087 

8.768 

0.56596 

11.904 

83 

0.83108 

2.831 

0.77293 

4-057 

0.70774 

5-647 

0.63641 

7-733 

0.56005 

10.526 

84 

0.82957 

2.436 

0.77079 

3-493 

0.70483 

4.867 

0.63252 

6.674 

0.55487 

9.105 

85 

0.82828 

2.037 

0.76898 

2.922 

0.70236 

4-075 

0.62921 

5-595 

0.55046 

7.647 

86 

0.82722 

1.634 

0.76750 

2-345 

0.70033 

3.272 

0.62648 

4.498 

0.54683 

6.157 

87 

0.82640 

1.228 

0.76634 

1.763 

0.69875 

2.462 

0.62435 

3.386 

0.54399 

4.642 

88 

0.82581 

0.820 

0.76551 

1.177 

0.69761 

1.645 

0.62283 

2.263 

0.54195 

3.106 

89 

0.82546 

0.410 

0.76501 

0.589 

0.69693 

0.823 

0.62192 

I-I33 

0.54072 

i-SS^ 

9° 

0.82534 

o.ooo 

0.76484 

o.ooo 

0.69671 

o.ooo 

0.62161 

o.ooo 

0.54030 

o.ooo 

Examples,     cosh  (0.7  771°)  =  0.82166  /g°.goo  =  0.82166  /g° 
cosh"1  (0.69857  /24°.9ii)  =  1.0/69°. 


[9] 


TABLE  II.    HYPERBOLIC   COSINES,     cosh  (p  /5)  =  r  /jy.     CONTINUED 


1.4 


45 
46 

47 
48 
49 

I-H57 
1.0959 

1-0759 
1-0559 
J-0359 

32.686 
33-o67 
33-424 
33-754 
34-056 

I.  1608 
1.1376 
I.  1143 

1.0910 
1.0676 

38.076 
38.582 

39.063 
39-517 

39-943 

1.2163 

1.1897 
1.1630 

1.1363 
1.1095 

43-570 

44.210 

44.827 
45-421 
45.989 

1.2830 
1.2529 

.2227 
.1926 
.1624 

49.084 
49.864 
50-625 

51-365 
52.083 

1.3616 

1-3279 
1.2941 

1.2604 
1.2267 

54-550 
55-471 
56.378 
57-266 
58.134 

50 
Si 
52 
53 
54 

1.0158 
0.99578 

0-97577 
0.95580 
0.93589 

34.328 
34-570 
34-780 

34-957 
35-!00 

1.0443 

1.0209 

0.99757 

0.97428 
0.95104 

40.341 
40.709 
41.045 

41-349 
41.618 

1.0828 
1.0561 
1.0294 
1.0027 

0.97613 

46.529 
47.041 

47-525 
47-978 

48.400 

.1322 

.IO2I 
.072O 
.O42I 
I.OI22 

52-777 
53-446 
54.090 
54.708 
55-299 

1.1931 
1.1596 

1.1262 
1.0929 

1.0598 

58.982 
59.8lO 
6o.6l8 
61.407 
62.175 

55 
56 
57 
58 
59 

0.91605 
0.89630 
0.87666 
0.85716 
0.83781 

35.206 
35-275 
35-305 
35-^93 
35-238 

0.92788 

0.90481 

0.88186 
0.85902 

0.83633 

41.850 
42.045 
42.201 
42.314 
42.383 

0.94964 
0.92325 
0.89697 
0.87082 

0.84482 

48.788 
49.141 
49.458 
49.736 

49-973 

0.98242 
0.95277 
0.92327 

0.89393 
0.86474 

55-862 
56-396 
56.900 
57-372 
57-Sii 

1.0268 

0.99406 
0.96149 
0.92912 
0.89696 

62.921 
63.645 
64-347 
65.025 
65.681 

60 
61 
62 

63 

64 

0.81862 
0.79961 
0.78081 
0.76223 
0.74390 

35.I39 
34.992 
34-794 
34-545 
34.242 

0.81379 
0.79144 

0.76929 

0.74735 
0.72565 

42.406 

42.379 
42.299 
42.164 
41.970 

0.81898 

0.79331 
0.76783 

0.74256 
0.71752 

50.168 

50-315 
50.412 

50.457 
50.446 

0-83573 
0.80692 
0.77831 
0.74991 
0.72173 

58.214 
58.578 
58.902 
59-185 
59.421 

0.86502 

0.83332 
0.80186 
0.77065 
0.73969 

66.313 
66.919 
67.497 
68.048 
68.571 

65 
66 

67 
68 

69 

0.72584 
0.70808 
0.69063 
0.67352 
0.65677 

33-881 
33-459 
32.974 
32-423 
31-804 

0.70420 
0.68304 
0.66218 

0.64164 
0.62146 

4i-7i3 
41.389 
40.993 
40.520 
39.966 

0.69271 

0.66815 
0.64385 

0.61985 
0.59618 

50-373 
50.235 
50.026 

49-739 
49.368 

0-69377 
0.66605 
0.63858 
0.61136 
0.58441 

59.608 
59-742 
59.8i7 
59.828 

59-769 

o.  70899 

0.67855 

0.64836 
0.61844 

0.58877 

69.064 
69.524 
69.950 
70.340 
70.689 

70 
7i 
72 
73 
74 

0.64043 
0.62452 
0.60907 
0.59410 
0.57966 

31.112 

30.344 
29.498 
28.570 
27-558 

0.60166 
0.58227 

0.56334 
0.54489 
0.52697 

39-324 
38.589 
37-754 
36.812 
35.756 

0.57283 
0.54984 
0.52724 
0.50506 
0.48335 

48.905 
48-342 
47.668 
46.870 
45-937 

0-55775 
0.53137 
0.50529 

0-47955 
0.45414 

59-633 
59.410 
59.090 
58-659 
58-104 

0-55937 
0.53022 
0.50134 
0.47271 
0-44434 

70.995 
7I-253 
7L458 
7I.6O2 
71.677 

75 
76 
77 
78 
79 

0.56579 
0-55252 
0-53990 
0.52796 
0-51675 

26.458 
25.269 
23.989 
22.616 
21.151 

0.50963 

0.49292 

0.47688 
0.46159 
0.44710 

34.578 
33-271 
31.828 
30.242 
28.507 

0.46215 

0.44151 

0.42149 
0.40216 
0.38360 

44.856 
43-6io 
42.180 
40.548 
38.692 

0.42910 
0.40446 
0.38028 
0.35658 
0-33345 

57405 
56.542 
55.487 
54.206 
52-659 

0.41622 
0.38836 
0.36075 

0-33339 
0.30630 

71.672 

71-573 
71.361 
71.015 
70.501 

80 
81 
82 

83 

84 

0.50631 
0.49669 
0.48794 
0.48009 
0.47320 

19-594 
17.946 
16.210 

14-391 
12.495 

0.43349 

0.42083 
0.40919 

0.39867 
0.38935 

26.616 
24-567 
22.359 
19-993 
17-475 

0.36591 
0.34921 
0.33362 
0.31928 
0.30635 

36-59I 
34-223 

3!-569 
28.608 

25-333 

0.31099 
0.28928 
0.26848 
0.24877 
0.23039 

50.800 
48.566 
45-888 
42.681 
38.853 

0.27951 
0.25304 
0.22691 

O.2OI22 
0.17608 

69.774 
68.774 
67-4I3 
65-561 
63.018 

85 
86 

87 
88 

89 

0.46730 
0.46241 

0-45857 
0.45581 

0.45415 

10.529 
8.501 
6.422 
4-304 
2-159 

0.38129 

0.37457 
0.36927 

0.36545 
0.36314 

14.813 

I2.O22 
9-I2I 
6.132 
3.081 

0.29499 
0.28538 

0.27770 
0.27208 

0.26865 

21.739 
17.838 
13-659 
9-247 
4.668 

0.21368 
0.19902 
0.18684 
0.17765 
0.17193 

34-303 
28.949 
22.746 

15-729 
8.051 

0.15169 
0.12853 
O.IO7IO 
0.08871 
0.07570 

59-474 
54.420 
47.029 
36-059 
20.198 

90 

0.45360 

o.ooo 

0.36236 

O.OOO 

0.26750 

o.ooo 

0.16997 

o.ooo 

0.07074 

o.ooo 

Examples,     cosh  (1.3  /73°)  =  0.50506  /46°.87o  =  0.50506  /46°.52'.i2" 
cosh"1  (0.07074  /o°)  =  1.5  /9Q°- 


[10] 


TABLE  II.  HYPERBOLIC  COSINES. 

1.6             1.7             i 

cosh  (p  ^S)  =  r  /y.  CONTINUED 
8           1.9           2.0 

0 

o 

o 

o 

0 

0 

45 

1-4524 

59.916 

I-5556 

65.149 

1.6714 

70.229 

1.7999 

75.I52 

I.94I3 

79.922 

46 

.4149 

60.974 

1.5141 

66.336 

1.6257 

71.536 

1.7496 

76.569 

1.8861 

81.437 

47 

•3774 

62.021 

1.4727 

67.516 

1.5802 

72.837 

1.6997 

77-979 

1-8313 

82.947 

48 

.3400 

63-051 

1.4316 

68.681 

1-5350 

74.125 

1.6502 

79-378 

1.7771 

84-445 

49 

.3028 

64.066 

1.3906 

69-833 

1.4901 

75.403 

i.  6012 

80.768 

1-7235 

85-935 

50 

.2657 

65.065 

1-3499 

70.973 

1.4456 

76.671 

1-5526 

82.149 

1.6706 

87.417 

.2287 

66.049 

I-3095 

72.IOO 

1.4015 

77.929 

1.5046 

83-522 

1.6184 

88.890 

52 

.1919 

67.018 

1.2693 

73-217 

1-3579 

79.179 

1-4572 

84.890 

1.5669 

90-356 

53 

•1554 

67.971 

1.2295 

74-324 

I.3I47 

80.422 

1.4104 

86.252 

1.5162 

91.817 

54 

.1191 

68.909 

1.1900 

75-421 

1.2719 

81.659 

1-3643 

87.609 

1.4664 

93-275 

55 

1.0831 

69-833 

1.1509 

76.508 

1.2297 

82.892 

1.3188 

88.963 

1.4174 

94.731 

56 

1-0473 

70.744 

I.II2I 

77.589 

1.1879 

84.122 

1.2740 

90.318 

1-3693 

96.188 

57 

1.0118 

71.640 

1.0737 

78.663 

1.1467 

85-350 

1.2299 

91.674 

1.3221 

97.646 

58 

0-97653 

72.522 

1-0358 

79.730 

1.1061 

86.576 

1.1865 

93.032 

1.2758 

99.108 

59 

0.94160 

73-392 

0.99826 

80.795 

i.  066  1 

87.808 

I-I439 

94-398 

1-2305 

100.577 

60 

0.90699 

74-249 

0.96117 

81.859 

1.0266 

89.045 

1.IO2O 

95-773 

1.1861 

102.057 

61 

0.87268 

75-094 

0.92451 

82.919 

0.98772 

90.287 

1.0609 

97.I58 

1.1427 

103.548 

62 

0.83871 

75.926 

0.88831 

83-982 

0.94946 

91.540 

I.O207 

98.559 

1.1003 

105.054 

63 

0.80508 

76.748 

0.85256 

85-048 

0.91182 

92.807 

0.98119 

99.978 

1.0589 

106.580 

64 

0.77177 

77o6i 

0.81727 

86.121 

0.87480 

94.092 

0.94254 

101.420 

1.0186 

108.129 

65 

0.73879 

78.365 

0.78245 

87.204 

0.83842 

95.400 

0.90472 

102.891 

0.97928 

109.707 

66 

0.70615 

79.161 

O.748IO 

88.300 

0.80268 

96.734 

0.86773 

104.394 

0.94100 

111.318 

67 

0.67383 

79-951 

0.71420 

89.415 

0.76759 

98.101 

O.86l6o 

105-937 

0.90378 

112.968 

68 

0.64184 

80.736 

0.68078 

90.551 

o.733i4 

99.508 

0.79631 

107.527 

0.86764 

114.663 

69 

0.61019 

81.519 

0.64782 

91.717 

0.69935 

100.962 

0.76190 

109.170 

0.83259 

116.410 

70 

0.57887 

82.300 

0.61533 

92.917 

0.66624 

102.473 

0.72838 

110.875 

0.79865 

118.215 

0.54786 

83.084 

0.58331 

94.161 

0.63379 

104.050 

0.69576 

112.653 

0.76583 

120.087 

72 

0-51715 

83.872 

0.55175 

95-458 

0.60202 

105.706 

0.66406 

114.514 

0.73414 

122.035 

73 

0.48674 

84.668 

0.52065 

96.819 

0.57095 

107.453 

0.63329 

116.471 

0.70361 

124.068 

74 

0.45663 

85-477 

0.49003 

98.258 

0.54061 

109.308 

0.60349 

"8-537 

0.67428 

126.197 

75 

0.42680 

86.304 

0.45988 

99-793 

0.51100 

111.291 

0.57470 

120.726 

0.64618 

128.432 

76 

0.39724 

87.156 

0.43021 

101.445 

0.48217 

"3-423 

0.54696 

123.058 

0.61934 

130.785 

77 

0.36794 

88.041 

0.40104 

103.241 

0.45415 

"5-732 

0.52030 

125.550 

0.59381 

133.267 

78 

0.33888 

88.972 

0.37238 

105.215 

0.42701 

118.248 

0.49480 

128.224 

0.56965 

135.890 

79 

0.31006 

89.963 

0.34428 

107.409 

0.40082 

121.008 

0.47055 

131.100 

0.54691 

138.664 

80 

0.28146 

91.036 

0.31679 

109.880 

0-37567 

124.055 

0.44764 

134.201 

0.52568 

141.601 

81 

0.25309 

92.221 

0.28997 

112.701 

0-35167 

127.438 

0.42617 

137-553 

0.50603 

144.713 

82 

0.22494 

93-56i 

0.26395 

"5-971 

0.32897 

131.210 

0.40626 

141.172 

0.48805 

148.003 

83 

0.19700 

95.128 

0.23887 

119.816 

0.30779 

135-433 

0.38806 

145.082 

0.47184 

151.474 

84 

0.16926 

97-033 

0.21497 

124.413 

0.28838 

140.167 

0.37174 

149.292 

0-4575° 

155-127 

85 

0.14177 

99-473 

O.I926I 

129.984 

0.27102 

145-463 

0-35749 

153.806 

0.44514 

158.954 

86 

0.11467 

102.829 

O.I723I 

136.806 

0.25603 

151-356 

0.34550 

158.614 

0.43486 

162.943 

87 

0.08809 

107.920 

0.15476 

145.176 

0.24378 

157-847 

0-33593 

163.691 

0.42674 

167.072 

88 

0.06261 

116.850 

0.14093 

I55-306 

0.23469 

164.877 

0.32895 

168.991 

0.42088 

171.315 

89 

0.04025 

136-057 

O.I3I95 

167.116 

0.22911 

172.324 

0.32472 

174-453 

0.41734 

175-637 

90 

0.02920 

180.000 

0.12884 

180.000 

0.22720 

180.000 

0.32329 

180.000 

0.41615 

180.000 

Examples,     cosh  (2.0  /go0)  =  0.41615  /i8o°. 

cosh"1  (0.54691  /i38°.664)  =  2.0  779°. 


TABLE  II.    HYPERBOLIC   COSINES,     cosh  (p  {8)  =  r  /_V     CONTINUED 


2.2 


2-3 


2.4 


2-5 


45 

2.0958 

84-55I 

2.2636 

89.050 

2-4449 

93.438 

2.6403 

97.730 

2.8502 

101.944 

46 

2.0350 

86.159 

2.1966 

90.740 

2.3711 

95.202 

2-5589 

99.564 

2.7603 

103.842 

47 

1-9749 

87-755 

2.1306 

92.417 

2.2984 

96.952 

2.4788 

101.380 

2.672O 

105.720 

48 

I-9I55 

89-341 

2-0655 

94.081 

2.2269 

98.687 

2.4001 

103.179 

2.5853 

107.578 

49 

1.8569 

90.917 

2.0013 

95-734 

2.1567 

100.408 

2.3229 

104.962 

2.5004 

109.417 

5° 

1.7992 

92.484 

1.9382 

97.376 

2.0877 

102.116 

2.2473 

106.729 

2.4173 

111.238 

51 

1.7424 

94.043 

1-8763 

99.009 

2.O2OO 

103.812 

2.1732 

108.482 

2.3361 

113.041 

52 

1.6865 

95-594 

I-8I55 

100.633 

1-9537 

105.498 

2.1008 

IIO.220 

2.2568 

114.826 

53 

1.6316 

97.140 

1-7559 

102.249 

1.8889 

107.173 

2.0301 

III.944 

2-1795 

"6.593 

54 

1-5777 

98.681 

1.6976 

103.858 

1-8255 

108.838 

1.9612 

113.656 

2.IO42 

118.344 

55 

I-5249 

IOO.22O 

1.6406 

105.462 

1.7637 

110.495 

1.8940 

"5.356 

2.0310 

120.079 

56 

1-4732 

101.758 

1.5848 

107.063 

1-7034 

112.145 

1.8286 

117.045 

1-9599 

121.799 

57 

1.4226 

103.296 

I-5303 

108.662 

1.6447 

113.789 

1.7651 

118.724 

1.8909 

123.504 

58 

i-373i 

104.836 

1.4772 

110.261 

1.5876 

115-430 

1-7035 

120-394 

1.8241 

125-195 

59 

1.3248 

106.382 

1.4256 

111.862 

1.5322 

117.069 

1.6438 

122.057 

1-7594 

126.874 

60 

1.2776 

107.936 

1-3754 

113.467 

1.4784 

118.708 

1-5859 

123.715 

1.6969 

128.541 

61 

1.2317 

109.501 

1.3266 

115.079 

1.4262 

120.347 

1.5299 

125.369 

1.6366 

130.198 

62 

1.1870 

III.O79 

1.2792 

116.700 

1.3758 

121.991 

1.4758 

I27.O20 

1.5785 

131.846 

63 

I-I435 

112.673 

1.2332 

118.333 

1.3270 

123.641 

1-4237 

128.671 

1.5226 

133.486 

64 

I.IOI2 

114.288 

1.1887 

119.982 

1.2799 

125.300 

I.3736 

130.322 

1.4689 

135.120 

65 

I.  O6O2 

115.928 

I-I457 

121.650 

1-2345 

126.970 

1-3254 

I3L977 

I.4I73 

136.749 

66 

I.02O4 

H7-597 

1.1042 

123-339 

1.1908 

128.654 

1.2791 

I33-638 

1.3679 

138.374 

67 

0.98193 

119.299 

1.0642 

125-055 

1.1488 

130.355 

1-2347 

I35-306 

1.3207 

139.998 

68 

0.94474 

I2I.O4I 

1.0256 

126.801 

1.  1086 

132.077 

1.1923 

136.985 

1.2756 

141.622 

69 

0.90886 

122.827 

0.98860 

128.581 

1.0700 

133-823 

1.1518 

138.676 

1.2327 

143-247 

70 

0.87429 

124.662 

0.95308 

130.400 

1-0332 

135-595 

1.1132 

140.382 

1.1919 

144.876 

7i 

0.84105 

126.554 

0.91908 

132.263 

0.99804 

137-397 

1.0765 

I42.IO6 

i-I532 

146.511 

72 

0.80915 

128.509 

0.88663 

134.174 

0.96464 

I39-233 

1.0417 

143.850 

1.1166 

148.153 

73 

0.77861 

130.533 

0-85573 

136.137 

0.93297 

141.106 

i.  0088 

I45.6l6 

1.0821 

149.803 

74 

Q-74945 

132.634 

0.82638 

138.158 

0.90303 

143.019 

0.97785 

147407 

1.0497 

151.464 

75 

0.72171 

134.818 

0.79862 

140.242 

0.87481 

144.976 

0.94877 

149.225 

1.0193 

I53-I37 

76 

0.69541 

137.094 

0.77246 

142.393 

0.84833 

146.980 

0.92156 

I5I.07I 

0.99093 

154-823 

77 

0.67060 

139.468 

0.74789 

144.615 

0.82360 

149-033 

0.89622 

152.948 

0.96459 

156.524 

78 

0.64732 

141.946 

0.72497 

146.911 

0.80062 

151-136 

0.87276 

I54-857 

0.94026 

158.241 

79 

0.62558 

144-534 

0.70372 

149.285 

0-77940 

153-291 

0.85117 

156.798 

0.91793 

159-974 

80 

0.60544 

147.237 

0.68416 

I5I-738 

0.75996 

155-499 

0-83145 

158.771 

0.89757 

161.723 

81 

0.58697 

150.058 

0.66632 

154-271 

0.74231 

157.760 

0.81361 

160.778 

0.87918 

163.490 

82 

0.57021 

152.997 

0.65024 

156.883 

0.72646 

160.074 

0.79764 

162.817 

0.86276 

165.274 

83 

0.55523 

156.054 

0.63595 

159-572 

0.71244 

162.438 

0-78354 

164.888 

0.84829 

167.074 

84 

0.54209 

I59-225 

0.62349 

162.334 

0.70025 

164.848 

0.77131 

166.987 

0-83575 

168.889 

85 

0-53085 

162.503 

0.61287 

165.164 

0.68990 

167.301 

0.76095 

I69.II2 

0.82518 

170.717 

86 

0.52156 

165.876 

0.60413 

168.054 

0.68141 

169.793 

0.75247 

171.260 

0.81650 

172-558 

87 

0.51428 

169.329 

0-59731 

170.994 

0.67479 

172.316 

0.74587 

I73.427 

0.80979 

174.409 

88 

0-50905 

172.849 

0.59242 

173-973 

0.67006 

174.863 

0.74116 

I75.6IO 

0.80499 

176.269 

89 

0.50590 

176.413 

0.58948 

176.980 

0.66722 

177.427 

0.73834 

177.803 

0.80207 

178.134 

90 

0.50485 

iSo.OOO 

0.58850 

180.000 

0.66628 

180.000 

0.73740 

iSo.OOO 

0.80114 

180.000 

Examples,     cosh  (2.2  745°)  =  2.2636  /89°.O5o  =  2.2636  /89°.o3'.oo<'. 
cosh"1  (1.0821  /I49°.8o3)  =  2.5  773°. 


[12] 


TABLE  II.     HYPERBOLIC   COSINES,     cosh  (p  /5)  =  r  /T.     CONTINUED 


2.6 


2.7 


2.9 


45 

3-°753 

106.093 

3-3I63 

110.190 

3-5741 

114.248 

3-8497 

118.275 

4-1443 

122.282 

46 

2.9758 

I08.O5I 

3.2O62 

112.207 

3-4523 

116.322 

3-7I46 

I  20.406 

3-9945 

124.469 

47 

2.8783 

109.987 

3-0984 

114.199 

3-3329 

118.368 

3-5826 

122.506 

3-8483 

126.623 

48 

2.7827 

III.9OI 

2.9928 

116.166 

3.2162 

'  120.387 

3-4536 

124-575 

3-7057 

128.743 

49 

2.6891 

113-793 

2.8896 

118.108 

3.1023 

122.378 

3.3278 

I26.6I4 

3-5667 

130.828 

5° 

2-5977 

115.663 

2.7889 

120.025 

2.9913 

124.340 

3-2053 

128.621 

3-43J5 

132.879 

5i 

2.5085 

II7.5I2 

2.6908 

121.917 

2.8832 

126.273 

3.0861 

130.595 

3.3001 

134.894 

52 

2.4216 

119.340 

2-5952 

123.784 

2.7780 

128.177 

2.9703 

132.536 

3.1726  ' 

136.873 

53 

2.3369 

I2I.I47 

2.5023 

125.626 

2.6759 

130.052 

2.8580 

134-444 

3.0490 

I38.8I5 

54 

2-2545 

122.931 

2.4I2O 

127.442 

2.5769 

131.898 

2.7492 

136.320 

2-9294 

I4O.72O 

55 

2.I74S 

124.695 

2.3245 

129.233 

2.4809 

I33.7I5 

2.6439 

138.162 

2.8137 

142.587 

56 

2.0970 

I26.44O 

2.2397 

131.000 

2.3880 

I35-502 

2.5421 

139.969 

2.7020 

I44-4I5 

57 

2.0219 

128.165 

2.1577 

132.742 

2.2983 

I37-259 

2-4439 

141.742 

2-5943 

146.203 

58 

1.9492 

129.872 

2.0785 

134.460 

2.  2Il8 

138.987 

2.3492 

143-479 

2.4905 

I47-95I 

59 

1.8790 

131.560 

2.0021 

136-154 

2.1284 

140.685 

2.2580 

145.180 

2.3907 

149.659 

60 

1.8113 

I33-23I 

1.9284 

137.824 

2.0481 

142.354 

2.1703 

146.846 

2.2949 

I5I-32S 

61 

1.7460 

134.885 

1.8575 

I39.47I 

1.9709 

143-993 

2.0861 

148.476 

2.2030 

152.949 

62 

1.6832 

136.523 

1.7894 

141.095 

1.8968 

145.601 

2.0053 

150.071 

2.1148 

I54-53Q 

63 

1.6229 

138.146 

I.724I 

142.696 

1.8259 

147.179 

1.9280 

151.627 

2.0304 

156.067 

64 

1.5650 

139-754 

I.66I5 

144.276 

1.7580 

148.728 

1.8541 

I53-I46 

1.9498 

I57-559 

65 

1.5096 

141.349 

1.  6oi6 

I45-834 

1.6931 

150.247 

1-7835 

154.628 

1.8729 

159.005 

66 

1.4566 

142.932 

1-5445 

I47-37I 

I.63II 

I5L737 

1.7162 

156.071 

1.7997 

160.406 

67 

1.4060 

144.504 

1.4900 

148.887 

I.572I 

I53.I98 

1.6522 

157-477 

1.7301 

161.759 

68 

1-3578 

146.066 

1.4382 

150-384 

I.5l6l 

154.628 

I-59I4 

158.844 

1.6640 

163.064 

69 

1.3120 

I47.62O 

1.3889 

151.863 

1.4629 

156.030 

1-5338 

I6O.I72 

1.6013 

164.321 

70 

1.2685 

149.164 

1.3422 

I53-322 

I.4I26 

157.404 

1-4793 

161.460 

1.5420 

165.528 

7i 

1.2273 

I50-705 

1.2981 

154.764 

1.3650 

158.749 

1.4278 

162.709 

1.4861 

166.685 

72 

1.1885 

152.239 

1-2565 

156.189 

1.3202 

160.066 

1-3793 

163.919 

1-4335 

167.792 

73 

i-^Q 

I53-770 

1.2174 

157-599 

1.2782 

I6I.355 

1.3338 

165.090 

1.3840 

168.848 

74 

1.1176 

I55-298 

1.1807 

158.994 

1.2388 

162.617 

1.2912 

166.222 

1-3377 

169.853 

75 

1.0855 

156.825 

1.1465 

160.374 

I.2O2O 

163.853 

1.2514 

167.315 

1.2946 

170.806 

76 

1.0556 

I58-35I 

1.1147 

161.742 

I.I677 

165.063 

1.2144 

168.370 

1.2545 

171.708 

77 

1.0278 

159.878 

1.0852 

163.096 

I.I360 

166.248 

1.  1802 

169.387 

1.2174 

172.560 

78 

I.OO22 

161.406 

1.0580 

164.440 

1.1069 

167.409 

1.1486 

170.368 

1.1832 

173.362 

79 

0.97880 

162.937 

1-0331 

165.773 

1  .0803 

168.547 

I.I2OO 

I7I-3I3 

1.1519 

174.116 

80 

0-95745 

164.471 

1.0105 

167.096 

1.0561 

169.664 

1.0938 

172.225 

J-^SS 

174.823 

81 

0.93821 

I66.008 

0.99006 

168.411 

1.0342 

170.760 

I.O7O2 

173.104 

1.0979 

175-485 

82 

0.92106 

167.549 

0.97189 

169.718 

I.OI47 

171.837 

I.O492 

173-953 

1.0751 

176.104 

83 

0.90596 

169.094 

0.95592 

171.018 

0.99765 

172.897 

1.0307 

174-774 

1-0550 

176.685 

84 

0.89290 

170.643 

0.94211 

172.312 

0.98287 

173.942 

I.OI48 

175-571 

1.0377 

177.229 

85 

0.88l87 

172.196 

0.93045 

173.601 

0.97041 

174-973 

I.OOI4 

176.344 

1.0231 

177.742 

86 

0.87286 

I73.752 

0.92093 

174.886 

0.96025 

175-993 

0.99042 

177.099 

I.OII2 

178.228 

87 

0.86587 

I75.3II 

0.91354 

176.168 

0.95235 

177.003 

0.98190 

177.838 

I.OOI9 

178.691 

88 

0.86089 

176.873 

0.90827 

177-447 

0.94671 

178.006 

0.97582 

178.566 

0.99528 

179.137 

89 

0.85789 

178.436 

0.90512 

178.724 

0-94334 

179.004 

0.97217 

I79-285 

0.99130 

I7.9-S7I 

90 

0.85689 

iSo.OOO 

0.96407 

180.000 

O.94222 

iSo.OOO 

0.97096 

iSo.OOO 

0.98999 

180.000 

Examples,     cosh  (2.8  785°)  =  0,97041  /I74°.g73  =  0.97041  /i74°.g8/.23* 
cosh"1  (i.5420/i65°.528)  =  3.0  /7Q°. 


[13] 


TABLE  III.    HYPERBOLIC  TANGENTS,     tanh  (p  [$)  =  r 


o-3 


0.4 


o-S 


45 
46 

47 
48 

49 

o.ioooo  44.812 
o.ioooi  45.812 

O.IO002  46.812 

0.10003  47.812 
0.10004  48.813 

0.19997  44-235 
0.20006  45.236 
0.20015  46.237 
0.20024  47-239 
0.20034  48.242 

0.29981  43.282 
0.30012  44.281 
0.30043  45.280 
0.30075  46.283 
0.30107  47.290 

0.39921  41-954 
0-39995  42.945 
0.40069  43.942 
0.40143  44-942 
0.40217  45-947 

0.49757  40.250 
0.49902  41.229 
0.50047  42.219 
0.50192  43.213 
0.50340  44.213 

50 
Si 

52 

53 
54 

0.10006  49.813 
0.10007  50.813 
0.10008  51.814 
0.10009  52.816 
o.iooio  53.818 

0.20043  49-245 
0.20053  50.251 
0.20062  51.256 
0.20071  52.262 
0.20081  53.268 

0.30138  48-297 
0.30169  49.306 
0.30201  50.315 
0.30232  51.328 
0.30263  52.344 

0.40293  46.955 
0.40370  47-969 
0.40446  48.986 
•  0.40521  50.005 
0.40596  51.031 

0.50490  45.220 
0.50639  46.233 
0.50789  47-252 
0.50939  48.276 
0.51089  49.308 

55 
56 
57 
58 
59 

o.ioon  54.820 

0.10013  55.823 

0.10014  56.825 
0.10015  57.828 
0.10016  58.830 

0.20090  54.276 
0.20099  55-285 
0.20107  56.295 
0.20115  57-306 
0.20124  58.318 

0.30294  53.362 
0-30325  54-382 
0-30355  55-404 
0-30385  56.426 
0,30414  57.452 

0.40671  52.058 
0.40746  53.089 
0.40820  54.124 
0.40892  55.164 
0.40964  56.207 

0.51239  50.344 
0.51389  51.388 
0-51538  52.439 
0.51687  53.494 
0.51835  54.556 

6o 
61 
62 
63 
64 

0.10017  59-833 

0.10018  60.837 
0.10019  61.841 

O.I002O  62.845 
O.IOO2O  63.849 

0.20132  59.332 
0.20140  60.345 
0.20148  61.360 
0.20156  62.375 
0.20164  63.391 

0.30444  58.479 

0.30473  59-5Io 
0.30501  60.542 
0.30528  61.575 
0-30555  62.610 

0.41037  57-255 
0.41107  58.305 
0.41176  59-359 
0.41244  60.417 
0.41311  61.479 

0-51983  55-625 
0.52128  56.700 
0.52271  57.781 
0.52412  58.867 
0-52552  59-959 

65 
66 

67 
68 

69 

O.IOO2I  64.853 
0.10022  65.857 
O.IOO23  66.862 
O.IOO24  67.866 
O.IOO25  68.871 

0.20171  64.408 
0.20179  65.425 
0.20186  66.443 
0.20193  67.461 
0.20199  68.480 

0.30580  63.647 
0.30605  64.686 
0.30630  65.727 
0.30655  66.769 
0.30678  67.813 

0.41376  62.544 
0.41441  63.612 
0.41504  64.685 
0.41565  65.760 
0.41623  66.837 

0.52689  61.058 
0.52824  62.164 
0.52957  63.276 
0.53085  64.391 
0.53209  65.511 

70 

72 
73 
74 

O.IOO26  69.877 
0.10026  70.882 
0.10027  71.888 
0.10028  72.893 
0.10028  73.898 

0.20206  69.501 

O.2O2I2  7O.522 
0.20217  71.544 
O.2O223  72.566 
O.2O228  73.589 

0.30701  68.859 
0.30722  69.906 
0.30742  70-955 
0.30762  72.005 
0.30781  73-056 

0.41680  67.918 
0.41735  69.002 
0.41788  70.089 
0.41838  71,179 
0.41886  72.272 

0-53330  66.638 
0.53446  67.771 
0.53560  68.909 
0-53669  70-052 
0-53773  71-200 

75 
76 
77 
78 
79 

O.IOO29  74.904 

0.10029  75-910 
0.10030  76.916 
0.10030  77.922 
0.10031  78.928 

O.2O234  74.612 
0.20239  75.635 
0.20243  76.659 
0.20246  77.684 
0.2O25O  78.709 

0.30799  74.108 
0.30816  75.162 
0.30831  76.217 
0.30846  77-273 
0.30860  78.329 

0.41932  73.368 
0.41975  74.466 
0.42016  75.566 
0.42054  76.668 
0.42088  77.771 

0.53872  72.351 

0-53965  73-507 
0.54054  74-667 
0.54138  75.831 
0.54215  76.999 

80 
81 
82 

83 
84 

0.10031  79-934 
0.10032  80.940 
0.10032  81.946 
0.10032  82.953 
0.10033  83.960 

0.20254  79-734 
0.20257  80.759 
0.20260  81.785 
0.20263  82.812 
0.20265  83.838 

0.30872  79-386 
0.30884  80.445 
0.30894  81.506 
0.30904  82.567 
0.30912  83.628 

0.42120  78.876 
0.42150  79-983 
0.42177  81.092 
0.42201  82.203 
0.42222  83.315 

0.54285  78.170 

0-54349  79-344 
0.54407  80.520 
0.54459  81.699 
0.54503  82.881 

85 
86 

87 
88 
89- 

0.10033  84.967 
0.10033  85.974 
0.10033  86.981 
0.10033  87.988 
0.10033  88.994 

0.20267  84.864 
0.20268  85.891 
0.20270  86.918 
0.20270  87.946 
0.20271  88.973 

0.30920  84.689 
0.30926  85.751 
0.30930  86.813 
0.30933  87.875 
0.30934  88.937 

0.42239  84.427 
0.42252  85.540 
0.42264  86.654 
0.42274  87.768 
0.42279  88.883 

0.54540  84.065 
0.54571  85.251 
0.54596  86.438 
0.54616  87.626 
0.54628  88.813 

90 

0.10033  90.000 

0.20271  90.000 

0.30934  90.000 

0.42280  90.000 

0.54631  90.000 

Note. 

tanh  (o  /J)  =  o  /j 

Examples. 

tanh  (0.5  /6o°)  = 

0.51983  /55°.625 

=  0.51983  /55°-37'. 

,?o". 

[14] 


TABLE  III.    HYPERBOLIC  TANGENTS,     tanh  (p    5)  =  r  /j.     CONTINUED 


0.6 


0.7 


0.8 


0.9 


45 

0.59406  38.183 

0.68732  35.786 

0-77577  33-098 

0-85756  30.161 

0.93077  27.044 

46 

0.59650  39.146 

0.69109  36.719 

0.78117  33.980 

0.86480  30.980 

0.93909  27.784 

47 

0.50898  40.119 

0.69495  37.663 

0.78671  34.878 

0.87229  31.815 

0.94950  28.539 

48 

0.60149  4I-°99 

0.69888  38.617 

0.79240  35.790 

0.88001  32.666 

0.95938  29.308 

49 

0.60403  42.088 

0.70287  39.581 

0.79824  36.715 

0.88799  33-531 

0.96966  30.092 

5° 

0.60660  43.085 

0.70694  40-557 

0.80421  37.653 

0.89620  34.412 

0.98032  30.892 

5i 

0.60920  44.092 

0.71107  41-545 

0.81031  38.605 

0.90466  35.310 

0.99136  31.710 

52 

0.61182  45.107 

0.71527  42.543 

0-81655  39-573 

0.9*337  36.223 

1.00282  32.545 

53 

0.61447  46.130 

0.71952  43-555 

0.82291  40.556 

0.92231  37.155 

1.01469  33.400 

54 

0.61713  47.162 

0.72382  44-577 

0.82940  41.554 

0.93150  38.105 

1.02697  34-274 

55 

0.61980  48.203 

0.72817  45.612 

0.83601  42.568 

0.94094  39.075 

1.03970  35.172 

56 

0.62248  49.254 

0-73257  46.662 

0.84274  43.599 

0.95063  40.066 

1.05287  36.091 

57 

0.62517  50.315 

0.73700  47-725 

0.84957  44-647 

0.96056  41.078 

1.06648  37-035 

58 

0.62785  51.384 

0.74145  48.800 

0.85649  45.712 

0.97069  42.111 

1.08054  38.004 

59 

0-63053  52-463 

0.74593  49-888 

0.86351  46.797 

0.98106  43.167 

1.09506  39.002 

60 

0.63322  53-552 

0.75047  50.990 

0.87063  47.900 

0.99168  44.247 

1.11009  40.026 

61 

0.63588  54-650 

0-75499  52-107 

0.87781  49.022 

1.00251  45-353 

1.12555  41.08,2 

62 

0-63852  55-758 

0-75950  53-238 

0.88504  50.165 

1.01353  46.486 

1.14144  42.168 

63 

0.64115  56.876 

0.76400  54.383 

0.89232  51.327 

1.02471  47-645 

I-I5777  43-289 

64 

0.64376  58.002 

0.76848  55.542 

0.89965  52.509 

1.03604  48.831 

I-I7454  44-445 

65 

0.64633  59.138 

0.77294  56.716 

0.90698  53.712 

1-04753  50.044 

1.19173  45-638 

66 

0.64886  60.284 

0-77737  57-902 

0.9U33  54.936 

1.05916  51.289 

1.20935  46.869 

67 

0-65135  61.439 

0.78177  59.105 

0.92166  56.180 

1.07090  52.562 

1.22737  48.140 

68 

0.65380  62.603 

0.78609  60.322 

0.92894  57.448 

1.08268  53.868 

1.24569  49-455 

69 

0.65618  63.777 

0.79033  61.553 

0.93616  58.737 

1.09447  55.204 

1.26429  50.812 

70 

0.65850  64.959 

0.79450  62.798 

0.94332  60.047 

1.10627  56.572 

1.28316  52.215 

?i 

0.66075  66.150 

0.79858  64.057 

0.95037  61.379 

1.11803  57.974 

1.30221  53.666 

72 

0.66294  67.350 

0.80256  65.329 

0.95727  62.732 

1.12972  59.408 

1.32140  55.164 

73 

0.66505  68.559 

0.80641  66.614 

0.96402  64.106 

1.14126  60.874 

1.34063  56.710 

74 

0.66708  69.775 

0.81014  67.913 

0.97060  65.501 

1.15260  62.372 

I-35986  58-307 

75 

0.66902  70.998 

0.81371  69.225 

0.97697  66.917 

1.16370  63.903 

1.37894  59-957 

76 

0.67086  72.228 

0.81713  70.550 

0.98311  68.352 

1.17450  65.468 

1-39775  61.658 

77 

0.67261  73.466 

0.82038  71.886 

0.98898  69.808 

1.18493  67.064 

1.41620  63.409 

78 

0.67425  74-7" 

0-82345  73.233 

0-99455  71-282 

1.19495  68.691 

1.43412  65.212 

79 

0.67577  75.962 

0.82632  74.591 

0.99981  72.773 

1.20447  70.347 

1.45141  67.065 

80 

0.67718  77.219 

0.82899  75.958 

1.00473  74.282 

1.21344  72.034 

1.46790  68.968 

81 

0.67847  78.481 

0.83144  77.334 

1.00926  75.805 

1.22179  73-746 

1.48345  70-919 

82 

0.67964  79.749 

0.83366  78.719 

I-OI339  77-344 

1.22946  75.483 

1.49790  72.913 

83 

0.68068  81.021 

0.83564  80.112 

1.01710  78.896 

1.23640  77-243 

1.51110  74-948 

84 

0.68159  82.296 

0.83738  81.512 

1.02036  80.460 

1.24253  79-025 

1.52289  77-023 

85 

0.68236  83.575 

0.83887  82.917 

1.02316  82.034 

1.24781  80.825 

I-533H  79.I32 

86 

0.68299  84.856 

0.84009  84.328 

1.02548  83.617 

1.25219  82.640 

1.54170  81.269 

87 

0.68349  86.140 

0.84105  85.743 

1.02730  85.206 

1.25566  84.468 

1.54848  83.429 

88 

0.68385  87.426 

0.84173  87.161 

1.02860  86.801 

1.25814  86.308 

1-55339  85.609 

89 

0.68406  88.713 

0.84214  88.580 

1.02937  88.400 

1.25963  88.153 

I-55637  87.802 

90 

0.68413  90.000 

0.84229  90.000 

1.02960  90.000 

1.26015  90.000 

1-55740  90-000 

Eaxmples. 

tanh  (0.9  /77°) 

=  1.18493  /67°.o64 

=  1.18493  /67°-03'. 

50*. 

(0.66708 


=  Q.6  /74°. 


[15] 


TABLE  III.    HYPERBOLIC  TANGENTS,     tanh  (p  IS)  =  r  £y.     CONTINUED 


i.i 


1.2 


1.4 


45 

0.99389 

23-833 

1.0457 

20.616 

1.0857 

17.464 

1.  1143 

14.484 

1-1323 

11.712 

46 

1.0049 

24.476 

1.0582 

21.144 

1.0993 

17.882 

1.1284 

14-775 

1.1464 

11.884 

47 

1.0164 

25-131 

1.0713 

21.685 

1.1136 

18.301 

1.1433 

15.064 

1.1613 

12.048 

48 

1.0283 

25-799 

1.0851 

22.236 

1.1286 

18.721 

1.1590 

15.352 

1.1770 

12.208 

49 

1.0408 

26.480 

1.0995 

22.798 

I-I445 

19.148 

1.1756 

15.640 

1.1936 

12.362 

5° 

1-0539 

27.178 

1.1147 

23-371 

1.1613 

19.584 

1.1932 

I5-930 

1.2112 

12.510 

1.0676 

27.891 

1.1307 

23-957 

1.1789 

20.027 

1.2118 

16.222 

1.2299 

12.652 

52 

1.0818 

28.621 

1.1474 

24-558 

1.1976 

20.479 

1-2315 

16.515 

1.2498 

12.787 

S3 

1.0967 

29.370 

1.1650 

25-I74 

1.2173 

20.941 

1.2524 

16.811 

1.2709 

12.914 

54 

1.1122 

30.139 

1.1835 

25.807 

1.2381 

21.414 

1.2746 

17.110 

1-2933 

13-035 

55 

I.I284 

30.930 

1.2030 

26.460 

1.2602 

21.900 

1.2982 

I7-4I3 

I.3I72 

13-151 

56 

I-I453 

3J-744 

1.2235 

27-I33 

1.2836 

22.401 

1.3234 

17.721 

L3427 

13.262 

57 

1.1629 

32-583 

1.2450 

27.827 

1.3084 

22.918 

1.3502 

18.036 

1.3700 

13.366 

58 

1.1813 

33-449 

1.2677 

28.546 

I.3346 

23-452 

1.3787 

18.358 

1.3992 

13.466 

59 

1.2005 

34-343 

1.2916 

29.292 

1.3626 

24.007 

1.4093 

18.689 

I-430S 

13-559 

6o 

1.2206 

35-267 

.3168 

30.068 

I.3923 

24.584 

1.4421 

19.032 

1.4642 

13-651 

<6i 

1.2415 

36.226 

•3434 

30.876 

1-4239 

25.187 

1.4772 

19.389 

I-5005 

13-738 

62 

1.2632 

37.222 

•3714 

31.720 

I.4576 

25.820 

1.5148 

19.761 

I-5396 

13.824 

63 

1.2858 

38-255 

.4009 

32.603 

1-4935 

26.485 

1-5553 

20.150 

1.5819 

13.908 

64 

1.3094 

39.328 

.4321 

33-527 

I-53I9 

27.186 

1.5990 

20.562 

1.6277 

i3-99i 

65 

1-3339 

40.446 

1.4649 

34.498 

I-5729 

27.928 

1.6463 

20.999 

1.6776 

14.076 

66 

1-3593 

41.612 

1-4995 

35-520 

1.6167 

28.715 

1.6974 

21.465 

I-73I9 

14.166 

67 

1.3856 

42.827 

1-5359 

36.597 

1.6637 

29-554 

1.7528 

21.966 

I.79H 

14.261 

68 

1.4129 

44.096 

1-5743 

37-737 

1.7140 

30.452 

1.8131 

22.507 

1.8566 

14.364 

69 

1.4411 

45.421 

1.6148 

38-941 

1.7680 

3I-4I5 

1.8788 

23.096 

1.9285 

14.480 

70 

1.4702 

46.806 

1.6573 

40.219 

1.8260 

32-452 

1.9506 

23.740 

2.0078 

14.612 

71 

1.5001 

48.256 

1.7020 

41-575 

1.8882 

33-570 

2.0293 

24.448 

2.0959 

14-765 

72 

1-5307 

49-773 

1.7488 

43-017 

I-955I 

34-782 

2.1158 

25.232 

2.1942 

14.944 

73 

1.5620 

51.361 

.1-7977 

44-553 

2.0269 

36.101 

2.2III 

26.106 

2.3043 

.15-159 

74 

I-5938 

53-022 

1.8488 

46.190 

2.1041 

37-539 

2.3164 

27.084 

2.4285 

15-419 

75 

1.6261 

54.762 

1.9019 

47-936 

2.1869 

39.110 

2-4332 

28.186 

2.5694 

15-734 

76 

1-6585 

56.581 

1.9568 

49.798 

2.2757 

40.830 

2.5631 

29-433 

2.7303 

16.120 

77 

1.6910 

58.482 

2.0134 

51-786 

2.3706 

42.720 

2.7078 

30-855 

2-9*57 

16.597 

78 

1.7232 

60.468 

2.0713 

53.906 

2.4716 

44-798 

2.8697 

32.486 

3-1312 

17.186 

79 

I-755° 

62.538 

2.1299 

56.165 

2.5786 

47.088 

3-0509 

34-367 

3-3843 

17.924 

80 

1.7858 

64.692 

2.1887 

58.571 

2.6912 

49.611 

3.2536 

36-546 

3-6845 

18.856 

81 

1-8155 

66.931 

2.2471 

61.126 

2.8085 

52-391 

3.4807 

39.085 

4.0459 

20.044 

82 

1.8435 

69.252 

2.3040 

63.832 

2.9290 

55-447 

3-7337 

42.056 

4.4876 

21.578 

83 

1.8696 

71.650 

2-3585 

66.690 

3.0506 

58.800 

4-0135 

45-546 

5-0365 

23-588 

84 

1-8933 

74.121 

2.4094 

69.694 

3-I704 

62.460 

4.3188 

49.646 

5-73" 

26.276 

85 

1.9142 

76.657 

2-4555 

72.836 

3-2845 

66.432 

4.6429 

54-459 

6.6297 

29-955 

86 

1.9318 

79.252 

2-4955 

76.102 

3-3884 

70.706 

4-9729 

60.069 

7.8015 

35-134 

87 

1.9460 

81.895 

2.5281 

79-475 

3-4767 

75-253 

5.2871 

66.523 

9-34I3 

42.644 

88 

1.9564 

84-575 

2.5523 

82.933 

3-5446 

80.030 

5-5531 

73-786 

11.259 

53-726 

89 

1.9627 

87.281 

2.5671 

86.452 

3-5875 

84.971 

5-7333 

81.707 

13.182 

69-695 

90 

1.9648 

90.000 

2.5721 

90.000 

3.6021 

90.000 

5.9978 

90.000 

14.101 

90.000 

Examples. 

tanh  (i 

4  /64°) 

=  1-5990 

/20°.562 

=  1.5990  /20°.33' 

43'- 

tanh-1  (1.7550  /62°.538  =  i.i  /79°. 


[16] 


TABLE  III.    HYPERBOLIC  TANGENTS,     tanh  (p  /8)  =  r  £/.     CONTINUED 


1.6 


1.8 


1.9 


45 

1.1413 

9.201 

1.1428 

6.984 

1-1385 

5-063 

.1302 

3-438 

1.1191 

2.094 

46 

1.1548 

9.267 

I.I553 

6.952 

1.1499 

4-950 

.1401 

3.260 

1.1275 

1.867 

47 

1.1691 

9.318 

1.1686 

6.902 

1.1618 

4.814 

•1505 

3-058 

1-1363 

1.613 

48 

1.1842 

9-358 

1.1826 

6.834 

I-I743 

4-659 

.1614 

2.832 

I-I455 

1-333 

49 

I.2OO2 

9-38S 

1.1974 

6-749 

1.1876 

4.480 

.1728 

2.578 

I-I550 

1-023 

So 

.2171 

9.400 

1.2131 

6.645 

1.2016 

4-275 

.1848 

2.295 

1.1650 

0.681 

•2351 

9.400 

1.2296 

6.520 

1.2163 

4-045 

•1975 

1.981 

I-I754 

0.306 

52 

•2541 

9-384 

1.2472 

6-373 

1.2318 

3-787 

.2107 

1.634 

1.1863 

0.103 

53 

.2744 

9-354 

1.2659 

6.203 

1.2483 

3-499 

•2247 

1-253 

1.1977 

0.550 

54 

.2960 

9-309 

1.2857 

6.008 

1.2657 

3.180 

•2394 

0-835 

1.2096 

1.039 

55 

.3190 

9.246 

1.3067 

5.788 

1.2842 

2.826 

•2549 

0-379 

I.222O 

I-57I 

56 

•3435 

9.166 

1-3292 

5-539 

1-3038 

2.436 

.2713 

O.I2I 

1.2350 

2.149 

57 

-3698 

9.069 

1-3532 

5.261 

1-3247 

2.008 

1.2886 

0.666 

1.2487 

2.777 

58 

•3979 

8-953 

1.3788 

4-953 

1.3469 

1.540 

1.3069 

1.259 

1.2630 

3.458 

59 

.4281 

8.817 

1.4063 

4.611 

1.3706 

1.026 

1.3263 

1.905 

1.2781 

4-195 

6o 

1.4606 

8.661 

I-4358 

4.232 

1-3959 

0.465 

1-3470 

2.606 

1.2939 

4-995 

61 

1.4956 

8.484 

I-4675 

3.819 

1-4230 

0.145 

1.3689 

3.366 

I.3I06 

5.860 

62 

1-5334 

8.286 

1.5017 

3.365 

1.4521 

0.810 

1.3922 

4.191 

I.328I 

6-793 

63 

1-5743 

8.065 

1.5386 

2.869 

1-4833 

1-533 

1.4170 

5-085 

1.3466 

7.802 

64 

1.6187 

7.819 

I-5785 

2.326 

1.5169 

2.318 

1-4435 

6.052 

1.3662 

8.891 

65 

1.6670 

7-548 

1.6218 

1-734 

I-553I 

3.172 

1.4718 

7.099 

1.3868 

10.068 

66 

1.7198 

7-252 

1.6690 

1.090 

i-59~23 

4.098 

1.5022 

8.232 

1.4086 

11-338 

67 

1.7777 

6.928 

1.7206 

0.387 

1-6347 

5.104 

1-5347 

9-459 

I-43I7 

12.707 

68 

1.8413 

6-575 

1.7770 

0.376 

1.6807 

6.196 

1.5696 

10.786 

1.4562 

14.184 

69 

1.9116 

6.191 

1-8389 

1.209 

1.7308 

7.382 

1.6070 

12.221 

1.4821 

I5.776 

70 

1.9893 

5-776 

1.9072 

2.116 

1-7854 

8.671 

1.6474 

13-773 

I-5095 

17.490 

2.0759 

5.326 

1.9828 

3.106 

1.8451 

10.074 

1.6908 

15-454 

I-5385 

19-337 

72 

2.1728 

4.840 

2.0667 

4.187 

1.9105 

11.602 

1.7376 

17-273 

I-5693 

21.326 

73 

2.2819 

4-3U 

2.1603 

5.371 

1.9823 

13.266 

1.7881 

19.244 

I.60I9 

23.466 

74 

2.4053 

3-744 

2.2652 

6.670 

2.0613 

15.084 

1.8425 

21.379 

1.6363 

25.769 

75 

2.5461 

3.127 

2.3834 

8.101 

2.1484 

17.076 

1.9011 

23.692 

1.6725 

28.245 

76 

2.7079 

2-457 

2.5172 

9-685 

2.2446 

19.261 

1.9641 

26.202 

I.7I05 

30-905 

77 

2.8957 

1.726 

2.6696 

11.447 

2.3510 

21.664 

2.0317 

28.925 

I-7503 

33-76o 

78 

3-IJ57 

0.922 

2.8442 

13-420 

2.4685 

24-3I5 

2.1040 

31.881 

I.79I7 

36.820 

79 

3.3766 

0.034 

3-0455 

15.644 

2.5984 

27.250 

2.1809 

35.088 

I-8344 

40.093 

80 

3.6909 

0.960 

3.2790 

18.175 

2.7416 

30.510 

2.2621 

38.568 

.    1.8781 

43-590 

81 

4-0755 

2.087 

3-55I9 

21.083 

2.8987 

34.140 

2.3469 

42.344 

1.9222 

47-319 

82 

4-5558 

3-390 

24.466 

3.0700 

38.191 

2-4344 

46.428 

1.9661 

51.281 

83 

5.1722 

4-937 

4.2487 

28.446 

3-2540 

42.722 

2.5230 

50.842 

2.OO9O 

55-474 

84 

5.9896 

6.839 

4.6927 

33-199 

3-4477 

47.793 

2.6104 

55.590 

2.0497 

59.894 

85 

7.1203 

9.289 

5.2102 

38.945 

3-6459 

53.448 

2.6940 

60.673 

2.0872 

64.528 

86 

8.7720 

12.668 

5-7994 

45-956 

3-8392 

59.720 

2.7701 

66.075 

2.I2O2 

69-359 

87 

11.388 

17.791 

6.4356 

54-527 

4-oi57 

66.606 

2.8350 

71.768 

2.1476 

74-359 

88 

I5-990 

26.760 

7-0503 

64.868 

4.1592 

74-043 

2.8849 

77.700 

2.1680 

79-494 

89 

24.844 

46.011 

7-5I93 

76.896 

4-253I 

81.905 

2.9163 

83.805 

2.1807 

84.723 

90       34.232   90.000      7.6968   90.000      4.2863   90.000      2.9271    90.000      2.1850  90.000 

Note.  Negative  quantities  are  in  heavy  type. 

Examples,     tanh  (1.6/54°)  =  1.2960  /9°.309  =  1.2960  /9°.i8'.32*. 
tanh  (2.0/64^)  =  1.3662  \8°.89i  =  1.3662  \8°.53'.28'. 
tr1  (1.4718  \7°.o99)  =  1.9/65°. 


TABLE  III.    HYPERBOLIC  TANGENTS,     tanh  (p  [S)  =  r  /jy.     CONTINUED 


2.3 


2.4 


2.5 


45 

1.1065 

1.007 

i 

.0932 

0-155 

1.0799 

0.492 

1.0672 

0.961 

1-0553 

1.283 

46 

1.1134 

0.746 

i 

.0987 

0.127 

1.0842 

0.783 

1.0702 

1.252 

1-0573 

1.564 

47 

1.  1206 

0.458 

i 

.1044 

0.436 

1.0884 

I.IOI 

1.0732 

1.567 

1.0592 

1.868 

48 

1.1280 

0.141 

i 

.1102 

0-774 

1.0927 

1.446 

1.0762 

1.908 

i.  0610 

2.195 

49 

I-I357 

0.206 

i 

.1161 

1.142 

1.0970 

1.820 

1.0790 

2.277 

1.0626 

2.548 

5° 

I-I437 

0.586 

i 

.1222 

1.542 

1.1013 

2.226 

1.0818 

2.676 

i  .0640 

2.929 

5i 

1.1519 

I.OOI 

i 

.1284 

1.978 

1.1056 

2.666 

1.0845 

3.107 

1-0653 

3-340 

52 

1.1604 

1.452 

i 

•1347 

2.452 

1.1099 

3-144 

1.0870 

3.572 

1.0663 

3-782 

S3 

1.1693 

1-943 

i 

.1411 

2.965 

1.1142 

3-659 

1.0894 

4.073 

1.0671 

4-256 

54 

1.1784 

2.476 

i 

•1477 

3-520 

1.1185 

4.215 

1.0917 

4.614 

1.0676 

4.766 

55 

1.1878 

3-055 

i 

-1544 

4.120 

1.1227 

4.815 

1.0937 

5.196 

1.0678 

5-314 

56 

1.1976 

3.682 

i 

.l6l2 

4.769 

1.1268 

5-462 

1-0955 

5.822 

1.0677 

5-903 

57 

1.2078 

4.361 

i 

.1681 

5-469 

1.1309 

6.160 

1.0971 

6.496 

1.0672 

6-534 

58 

1.2183 

5-094 

i 

-I75I 

6.226 

1.1348 

6.912 

1.0984 

7.220 

1.0662 

7.211 

59 

1.2292 

5.888 

i 

.1822 

7.042 

1.1386 

7.722 

1.0993 

7.998 

1.0647 

7.938 

60 

1.2405 

6-745 

i 

.1894 

7.921 

1.1422 

8-594 

1.0999 

8.835 

1.0628 

8.718 

61 

1.2522 

7.671 

i 

.1967 

8.869 

1-1457 

9-531 

I.IOOI 

9-735 

i  .0603 

9.556 

62 

1.2644 

8.669 

i 

.2041 

9.889 

1.1490 

10.540 

1.0999 

10.701 

1.0572 

10.455 

63 

1.2771 

9-744 

i 

.2Il6 

10.988 

1.1520 

11.625 

1.0992 

11.740 

1-0535 

11.419 

64 

1.2903 

10.902 

i 

.2192 

12.171 

1.1548 

12.791 

1  .0980 

12.855 

i  .0490 

12.455 

65 

I-3039 

12.151 

i 

.2268 

13-443 

1-1573 

14.044 

1.0962 

14-053 

1.0438 

13-567 

66 

1.3181 

13-496 

i 

-2344 

14.810 

1-1595 

15-390 

1.0939 

15-341 

1.0378 

14-761 

67 

I-3329 

14.942 

i 

.2421 

16.280 

1.1613 

16.836 

1.0909 

16.723 

1.0310 

16.044 

68 

1-3483 

16.499 

i 

.2499 

17-858 

1.1628 

18.389 

1.0872 

18.208 

1.0232 

17.424 

69 

1.3642 

18.172 

i 

.2576 

I9-552 

1.1639 

20.056 

1.0829 

19.802 

1.0145 

18.906 

70 

1.3808 

19.968 

i 

.2654 

21.370 

1.1645 

21.843 

1.0779 

21.514 

1.0049 

20.500 

7i 

1.3980 

21.898 

i 

.2732 

23-319 

1.1647 

23-759 

1.0721 

23-352 

0.99427 

22.215 

72 

I-4I59 

23.968 

i 

.2809 

25-405 

1.1644 

25.811 

1-0655 

25-324 

0.98265 

24.061 

73 

1-4344 

26.185 

i 

-2885 

27.636 

i-.i637 

28.006 

1.0582 

27-439 

0.97006 

26.048 

74 

1-4535 

28.560 

i 

.2961 

30.020 

1.1625 

30.352 

1.0501 

29.706 

0.95649 

28.187 

75 

1-4732 

31.099 

i 

.3036 

32.564 

1.  1608 

32.858 

1.0413 

32.134 

0.94202 

30.489 

76 

1-4934 

33-8II 

i 

.3109 

35-274 

1.1587 

35-531 

1.0318 

34-730 

0.92671 

32-966 

77 

1.5140 

36.702 

i 

.3l8l 

38.156 

1.1561 

38.374 

1.0218 

37-505 

0.91068 

35-633 

78 

I-5348 

39-778 

i 

•3251 

41.212 

I-I53I 

41.392 

1.0113 

40.467 

0.89405 

38.501 

79 

1-5559 

43-043 

1-3318 

44-446 

1.1498 

44.590 

1.0004 

43.620 

0.87701 

41-582 

80 

1.5768 

46.501 

i. 

.3383 

47-859 

1.1462 

47.968 

0.98931 

46.968 

0.85979 

44-887 

81 

1-5974 

50.151 

i 

•3444 

51-449 

1.1425 

5I.527 

0.97822 

50-517 

0.84264 

48.427 

82 

1.6174 

53.989 

i 

•35oo 

55-212 

1.1387 

55-264 

0.96733 

54-267 

0.82584 

52.209 

83 

1.6364 

58.011 

i, 

•3552 

59.140 

I-I349 

59.I7I 

0.95686 

58.212 

0.80977 

56.233 

84 

1.6541 

62.210 

i, 

•3599 

63.223 

1.1312 

63-237 

0.94710 

62.344 

0.79476 

60.499 

85 

1.6700 

66.570 

i, 

,3640 

67.449 

1.1279 

67-450 

0.93826 

66.650 

0.78120 

64.994 

86 

1.6837 

71.070 

i, 

3674 

71.800 

1.1250 

71-793 

0.93061 

71.113 

0.76946 

69.703 

87 

1.6949 

75.690 

i. 

37°2 

76.254 

1.1225 

76.245 

0.92439 

75-710 

0.75991 

74.598 

88 

1.7031 

80.406 

i, 

3722 

80.790 

1.1207 

80.781 

0.91979 

80.413 

0.75283 

79-645 

89 

1.7081 

85.187 

i. 

3734 

85.381 

1.1196 

85.376 

0.91696 

85.188 

0.74849 

84.796 

90 

1.7099 

90.000 

I-3738 

90.000 

1.1192 

90.000 

0.91601 

90.000 

0.74702 

90.000 

Note. 

Negative  quantities  are  in  heavy  type. 

Examples. 

tanh  (2.1  748°) 

=  1.1280  /o°.i4i 

=  1.1280 

/o°.o8'.28*. 

tanh  (2.1  /49°)  =  1.1357  \o°.2o6  =  1.1357  \o°  i2'.22*. 
(1.0318  \34°-73o)  =  2.4  776°. 
[18] 


TABLE  III.  HYPERBOLIC  TANGENTS,  tanh 
2.6           2.7           2.8 

(p  /$)  =  r  /y.  CONTINUED 
2.9          3.0 

0 

o 

o 

o 

0 

0 

45 

1.0445 

1.480 

1.0348 

1-576 

1.0264 

1-595 

1.0192 

1-554 

1.0131 

1.468 

46 

1.0456 

1.744 

I.0352 

I.82I 

1.0262 

1.816 

1.0185 

1.748 

I.OI2O 

1.637 

47 

1.0465 

2.030 

1-0354 

2.083 

1.0257 

2.051 

1.0175 

1-955 

I.OI07 

1.815 

48 

1-0473 

2.337 

1-0353 

2.365 

1.0250 

2.303 

1.0163 

2.175 

I.OOgi 

2.OO2 

49 

1.0479 

2.667 

1.0350 

2.666 

1.0240 

2.572 

1.0148 

2.409 

I.OO72 

2.198 

50 

1.0483 

3.022 

1.0344 

2.988 

1.0227 

2.857 

1.0129 

2.656 

I.OO49 

2.405 

51 

1.0483 

3.403 

1-0335 

3-332 

I.02IO 

3.160 

1.0106 

2.916 

1.0023 

2.622 

52 

1.0480 

3.812 

1.0323 

3.700 

I.OlSg 

3-483 

1.0080 

3-191 

0.99922 

2.849 

53 

1-0475 

4.250 

1.0307 

4.094 

1.0164 

3-826 

1.0049 

3.481 

0.99572 

3.086 

54 

1.0466 

4.717 

1.0286 

4-513 

i-OI35 

4.190 

1.0013 

3.788 

0.99174 

3-334 

55 

1.0452 

5.219 

I.O26O 

4-959 

I.OIOI 

4-577 

0.99722 

4.112 

0.98722 

3-593 

56 

1.0434 

5.756 

1.0230 

5-437 

1.  006  1 

4.987 

0.99254 

4-452 

0.98211 

3.862 

57 

1.0412 

6.331 

I.OI94 

5-945 

1.0015 

5-422 

0.98724 

4.810 

0.97637 

4.141 

58 

1-0385 

6.947 

I.OI52 

6.487 

0.99625 

5-884 

0.98126 

5-186 

0.96995 

4-432 

59 

1-0351 

7.606 

I.OIO4 

7.066 

0.99027 

6.374 

0-97454 

5-582 

0.96279 

4-735 

60 

1.0311 

8.313 

I.OO48 

7.684 

0-98353 

6.895 

0.96701 

5-999 

0.95481 

5-049 

61 

1.0265 

9.069 

0.99847 

8-343 

0-97597 

7-449 

0.95862 

6-439 

0-94597 

5-375 

62 

i.  02  1  1 

9.879 

0.99129 

9.048 

0.96751 

8.036 

0.94930 

6.904 

0.93619 

5-713 

63 

1.0149 

10.747 

0.98322 

9.801 

0.95807 

8.660 

0.93897 

7-392 

0.92540 

6.063 

64 

1.0079 

11.677 

O.9742O 

10.607 

0.94760 

9.325 

0.92756 

7.907 

0.91352 

6.426 

65 

0.99990 

12.675 

0.96414 

11.470 

0.93603 

10.033 

0.91500 

8.452 

0.90047 

6.801 

66 

0.99099 

13.747 

0.95300 

12.393 

0.92328 

10.789 

0.90121 

9.027 

0.88618 

7.192 

67 

0.98106 

14.898 

0.94070 

13-383 

0.90927 

H.599 

0.88612 

9.639 

0.87055 

7.598 

68 

0.97006 

16.136 

0.92720 

14.449 

0.89396 

12.464 

0.86964 

10.289 

0.85350 

8.021 

69 

0-95794 

17.469 

0.91243 

15-597 

0.87726 

13-396 

0.85169 

10.982 

0.83495 

8.463 

70 

0.94467 

18.902 

0.89635 

16.833 

0.85912 

14-399 

0.83221 

11.722 

0.81482 

8.926 

71 

0.93022 

20.451 

0.87893 

18.168 

0.83950 

15-482 

0.81113 

12.518 

0.79302 

9.414 

72 

0.91458 

22.122 

O.860I3 

19-613 

0.81834 

16.655 

0.78839 

I3.378 

0.76946 

9.932 

73 

0.89775 

23.929 

0.83994 

21.184 

0.79562 

I7.93I 

0.76394 

14-310 

0.74408 

10.485 

74 

0.87976 

25.884 

0.8l839 

22.893 

0.77134 

19-325 

0-73774 

15-328 

0.71683 

11.081 

75 

0.86066 

28.003 

0-79552 

24-757 

o.7455i 

20.857 

0.70977 

16.448 

0.68765 

11.729 

76 

0.84053 

30.302 

0.77140 

26.799 

0.71815 

22.547 

0.68005 

17.690 

0.65650 

12.443 

77 

0.81949 

32-799 

O.746I2 

29.038 

0.68938 

24:421 

0.64861 

19.078 

0.62338 

13-241 

78 

0.79769 

35.5I3 

0.71985 

31-505 

0.65930 

26.512 

0.61551 

20.647 

0.58829 

14.146 

79 

0-77535 

38.466 

0.69278 

34-231 

0.62807 

28.861 

0.58087 

22.437 

0-55129 

15.191 

80 

0.75271 

41.678 

O.6652O 

37-249 

0.59596 

3L5I9 

0.54486 

24.504 

0.51245 

16.419 

81 

0.73010 

45-171 

0.63743 

40.599 

0.56326 

34-543 

0.50773 

26.921 

0.47190 

17-893 

82 

0.70790 

48.966 

0.60990 

44-322 

0.53039 

38.004 

0.46981 

29.784 

0.42986 

19.700 

83 

0.68652 

53.078 

0.58311 

48.464 

0.49791 

41.989 

0.43156 

33-221 

0.38661 

21.976 

84 

0.66645 

57.5I7 

0.55765 

53.060 

0.46648 

46.591 

0-39363 

37.403 

0.34257 

24.916 

85 

0.64820 

62.286 

0.53421 

58.143 

0.43691 

5L907 

0.35689 

42.549 

0.29833 

28.839 

86 

0.63230 

67.370 

0.51353 

63.722 

0.41024 

58.025 

0-32255 

48.931 

0.25491 

34-254 

87 

0.61928 

72.740 

0.49636 

69.778 

0.38762 

64.988 

0.29220 

56.833 

0.21393 

41.990 

88 

0.60959 

78.349 

0.48347 

76.253 

0.37028 

72.762 

0.26791 

66.456 

0.17815 

53-294 

89 

0.60361 

84.129 

0-47544 

83.042 

0-35929 

81.195 

0.25198 

77.709 

0.15225 

69.480 

90 

0.60160 

9O.OOO 

0.47273 

90.000 

0-35553 

90.000 

0.24641 

90.000 

0.14255 

90.000 

Note.  Negative  quantites  are  in  heavy  type. 

Examples      tanh  (2.6  765°)  =  0.99990  /i2°.675  =  0.99990  /i2°.40/.3O*r. 
tanh"1  (0.88618  \7°.i92)  =  3.0/66°. 


[19] 


sinh  0 


1ABLE, 

1  V  .   ^UJX-IVC.^  . 

L  ±r\  \j  r  /\v^  i  u 

e   '' 

'1 

O.I 

0.2 

o-3 

0.4 

0-5 

0 

45 
46 
47 
48 
49 

o 

i.ooooo  0.096 

0-99993  o-°95 
0.99987  0.095 
0.99981  0.094 
0.99975  0.094 

o 
i.ooooo  0.383 

0.99975  0.382 
0.99950  0.381 
0.99930  0.380 
0.99910  0.378 

o 
i.ooooo  0.860 
0.99950  0.858 

0.99897  0.856 
0.99847  0.854 
0.99797  0.852 

o 

1.00013  1.532 
0.99920  1.529 

0.99828  1.526 
0-99733  1-520 
0.99640  1.513 

o 
1.00032  2.391 
0.99888  2.388 
0.99744  2.385 
0.99598  2.378 
0.99454  2.368 

So 
Si 

52 

53 

54 

0.99970  0.094 
0.99965  0.093 
0.99960  0.092 
0.99955  0.091 
0.99950  0.090 

0.99880  0.376 
0.99860  0.374 
0.99840  0.371 
0.99815  0.367 
0.99795  0.362 

0.99743  0.848 

0.99690  0.842 
0.99640  0.834 
0.99590  0.826 
0.99540  0.818 

0.99550  1.506 
0.99460  1.497 
0.99370  1.486 
0.99280  1.472 
0.99190  1.458 

0.99312  2.357 
0.99170  2.342 
0.99028  2.325 
0.98888  2.304 
0.98748  2.281 

55 
56 
57 
58 
59 

0.99944  0.089 
0.99939  0.088 
0-99933  0.086 
0.99928  0.085 
0.99922  0.083 

0-99775  0.357 
0.99755  0.352 
0.99730  0.347 
0.99705  0.342 
0.99685  0.336 

0.99490  0.809 
0.99443  0.799 
0.99397  0.787 
0.99347  0.773 
0.99300  0.760 

0.99103  1.440 
0.99018  1.421 
0.98930  1.400 
0.98845  1.378 
0.98763  1.354 

0.98610  2.254 
0.98476  2.226 
0.98344  2.195 
0.98212  2.l6o 
0.98082  2.123 

60 
61 
62 

63 

64 

0.99917  0.082 
0.99912  0.081 
0.99907  0.079 
0.99902  0.077 
0.99897  0.075 

0.99670  0.331 
0.99645  0.324 
0.99625  0.317 
0.99605  0.309 
0.99585  0.301 

0.99257  0.746 
0.99213  0.731 
0.99170  0.715 
0.99127  0.698 
0.99083  0.680 

0.98685  1.330 
0.98603  1.302 
0.98523  1.273 
0.98448  1.243 
0-98373  I-212 

0.97956  2.085 
0.97832  2.042 
0.97710  1.998 
0.97590  1.950 
0.97476  1.900 

65 
66 
67 
68 
69 

0.99893  0.073 
0.99889  0.071 
0.99885  0.069 
0.99881  0.066 
0.99877  0.064 

0.99570  0.293 
0-99555  0.284 
0.99540  0.275 
0.99520  0.265 
0.99505  0.255 

0.99040  0.661 
0.99000  0.641 
0.98963  0.621 
0.98927  0.599 
0.98891  0.577 

0.98300  1.179 
0.98232  1.144 
0.98165  1.108 
0.98100  1.070 
0.98035  1.030 

0.97364  1.848 
0.97254  1.794 
0.97150  1.738 
0.97046  1.679 
0.96946  1.617 

7o 

72 
73 

74 

0.99873  0.062 
0.99869  0.059 
0.99865  0.057 
0.99861  0.054 
0.99858  0.051 

0.99490  0.245 
0.99475  0.235 
0.99460  0.225 
0.99445  0.214 
0.99435  0.203 

0.98857  0.555 
0.98823  0.532 
0.98790  0.508 
0.98760  0.483 
0.98733  0.458 

0-97975  0.990 
0.97918  0.948 
0.97863  0.905 
0.97808  0.861 
0.97758  0.817 

0.96852  1.554 
0.96760  1.489 
0.96676  1.422 
0.96592  1.354 
0.96514  1.284 

75 
76 

77 
78 
79 

0.99855  0.048 
0.99852  0.045 
0.99850  0.042 
0.99847  0.039 
0.99845  0.036 

0.99425  0.192 
0.99415  0.180 
0.99405  0.168 
0-99395  0.156 
0.99385  0.144 

0.98707  0.432 
0.98680  0.406 
0.98653  0.379 
0.98633  0.351 
0.98613  0.322 

0.97710  0.771 
0.97665  0.724 
0.97625  0.676 
0.97585  0.628 
0-97545  0-578 

0.96440  1.  212 
0.96368  1.138 
0.96304  I.O62 
0.96246  0.986 
0.96190  0.909 

80 
81 
82 
83 
84 

0.99843  0.033 
0.99841  0.030 
0.99839  0.026 
0.99838  0.023 

0.99837  O.O2O 

0-99375  0.131 
0.99365  0.118 
0.99360  o.i  06 
0.99355  0.093 
0.99350  0.080 

0.98593  0.294 
0.98577  0.266 
0.98563  0.238 
0-98553  0.209 
0.98543  0.180 

0.97512  0.528 
0.97483  0.477 
0.97458  0.425 
0-97433  0.373 
0.97413  0.321 

0.96134  0.830 
0.96088  0.750 
0.96046  0.669 
0.96008  0.587 
0.95974  0.505 

85 
86 

87 
88 
89 

0.99836  O.OI7 
0.99835  0.014 

0.99834  o.on 

0.99833  0.008 

0.99832  0.004 

0-99345  0.067 
0.99340  0.054 
0-99335  0.041 
0-99335  0.028 
0-99335  0.014 

0.98537  0.150 

0.98530  O.I2O 
0,98523  0.090 
0.98517  O.66o 
0.98510  O.O3O 

0-97395  0.268 
0.97380  0.215 
0.97370  0.162 
0.97365  0.108 
0.97360  0.054 

0.95944  0.422 
0.95920  0.338 
0.95904  0.254 
0.95894  0.170 
0.95890  0.085 

90 

0.99831  o.ooo 

0-99335  o.ooo 

0.98507  o.ooo 

0-97355  o.ooo 

0.95886  o.ooo 

Note. 

sinh  0       h 

en  0  =  0/5.. 

—  i.o  wn 
6 

, 

Example. 

sinh  (0.3  769°) 

0.98891  /o°.577 

—  o  98891  /o°.34', 

,37'. 

Q-3  /69° 

[20] 


1AULK  J.V.   ^ 

UXS-IVC^  i  11\  \J  J 

6 

JNTINUED 

0 

45 
46 
47 
48 
49 

0.6 
0 

1.00070  3.440 
0.99863  3.437 

0.99655  3-434 
0.99445  3426 

0.99237  3-4I4 

0.7 

o 

1.00134  4.676 
0.99849  4.679 

0.99564  4.676 
0.99281  4  666 
0.98999  4.652 

0.8 
o 
1.00230  6.108 
0.99856  6.112 
0.99484  6.109 
0.99114  6.099 
0.98745  6.082 

0.9 

o 

1.00363  7.728 
0.99893  7.735 
0.99425  7.734 
0.98955  7.725 

0.98488  7.707 

1.0 

o 
1.00553  9-531 
0.99975  9.546 
0.90394  9-550 
0.98816  9.543 
0.98242  9.525 

50 

52 
53 
54 

0.99030  3.398 
0.98825  3.379 
0.98623  3.355 
0.98421  3.326 
0.98220  3.293 

0.98717  4.632 
0.98439  4.606 
0.98161  4-574 
0.97887  4-537 
0.97616  4.493 

0.98379  6.058 
0.98015  6.026 

0-97655  5-987 
0.97298  5.940 
0.96944  5.886 

0.98026  7.679 

0.97567  7-642 
0.97111  7.595 
0.96659  7.538 

0.96211  7.472 

0.97672  9-495 
0.97105  9.453 
0.96543  9.399 
0.95986  9.333 
0-95435  9-255 

55 
56 
57 
58 
59 

0.98023  3.256 
0.97830  3.215 
0.97638  3.171 
0.97448  3.122 
0.97262  3.069 

0.97349  4.445 
0.97084  4.391 
0.96823  4.331 
0.96564  4.265 
0.96313  4.193 

0.96594  5.824 
0.96249  5.755 
0-95909  5-678 
0-95574  5-593 
0.95244  5.502 

0.95769  7-396 

0-95333  7-312 
0.94904  7.218 
0.94482  7.114 
0.94067  7.000 

0.94890  9.166 

0-94353  9-065 
0.93825  8.952 
0.93305  8.827 
0.92795  8.691 

6o 
61 
62 
63 
64 

0.97081  3.013 
0.96903  2.953 
0.96728  2.889 
0.96557  2.821 
0.96390  2.749 

0.96067  4.117 
0.95826  4.036 
0.95589  3-951 
0.95356  3-859 

o-gs^o  3-762 

0.94923  5.405 
0.94608  5.300 
0.94299  5.189 
0.93996  5.070 
0.93703  4-944 

0.93661  6.878 
0.93263  6.747 
0.92874  6.607 
0.92493  6.458 
0.92121  6.300 

0.92295  8.544 
0.91805  8.385 
0.91325  8.215 
0.90856  8.033 
0.90400  7.841 

65 
66 
67 
68 
69 

0.96228  2.674 
0.96072  2.596 
0.95922  2.515 
0.95775  2.430 
0.95632  2.342 

0.94911  3.661 
0.94697  3.554 
0.94493  3.444 
0.94293  3.329 
0.94100  3.209 

0.93416  4.812 
0.93140  4.674 
0.92873  4.529 
0.92614  4.379 
0.92364  4.223 

0.91761-  6.136 
0.91411  5.962 
0.91074  5.780 
0.90747  5.590 
0.90430  5.392 

0.89957  7.637 
0.89527  7.424 
0.89111  7-2OI 
0.88708  6.968 
0.88320  6.723 

70 

72 
73 
74 

0.95495  2.251 
0.95365  2.157 
0.95242  2.061 
0.95123  1.962 
0.95010  1.860 

0.93914  3.085 
0.93737  2.957 
0.93569  2.825 
0.93407  2.689 
0.93254  2.550 

0.92123  4.061 
0.91891  3.894 
0.91671  3.722 
0.91461  3.544 
0.91261  3.361 

0.90127  5.187 
0.89837  4.975 
0.89558  4.756 
0.89292  4.530 
0.89041  4.298 

0.87947  6.469 
0.87589  6.207 
0.87247  5-936 
0.86921  5.656 
0.86612  5.368 

75 
76 

77 
78 
79 

0.94902  1.756 
0.94802  1.649 
0.94707  1.540 
0.94620  1.429 
0.94540  1.317 

0.93109  2.408 
0.92973  2.263 
0.92846  2.114 
0.92727  1.962 
0.92617  i.  808 

0.91073  3.174 
0.90895  2.982 
0.90729  2.787 
0.90574  2.588 
0.90431  2.385 

0.88804  4.059 
0.88581  3.815 
0.88372  3.566 
0.88177  3.312 
0-87997  3.053 

0.86320  5.072 
0.86045  4.769 
0.85788  4.458 
0.85549  4-I4I 
0.85328  3:818 

80 
81 
82 
83 
84 

0.94465  1.203 
0.94397  1.087 
0.94337  0.970 
0.94283  0.852 
0.94237  0.732 

0.92516  1.652 
0.92424  1.493 
0.92343  1.332 
0.92270  1.169 
0.92207  1.005 

0.90301  2.179 
0.90183  1.969 
0.90076  1.757 
0.89981  1.543 
0.89899  1.327 

0.87831  2.789 
0.87681  2.522 
0.87547  2.251 
0.87428  1.976 
0.87324  1.699 

0.85125  3.489 
0.84940  3.156 
0.84774  2.817 
0.84628  2.474 
0.84501  2.128 

85 
86 

87 
88 
89 

0.94197  0.612 
0.94163  0.490 
0.94138  0.368 
0.94122  0.246 
0.94110  0.123 

0.92153  0.839 
0.92109  0.673 
0.92074  0.506 
0.92050  0.338 
0.92036  0.169 

0.89829  1.109 
0.89771  0.889 
0.89726  0.668 
0.89694  0.446 
0.89675  0.223 

0.87237  1.420 
0.87164  1.138 
0.87108  0.855 
0.87068  0.571 
0.87043  0.286 

0.84393  1-779 
0.84305  1.426 
0.84236  1.071 
0.84186  0.715 
0.84156  0.358 

90 

0.94107  o.ooo 

0.92031  o.ooo 

0.89670  o.ooo 

0.87037  o.ooo 

0.84147  o.ooo 

•   E^p,, 

sinh  (1.0/85°) 

0.84393  /i°.779 

-0  84101/1°  46' 

.44". 

1.0/85° 

[21] 


sinh  d 


1ABI 
I.I 

,E  IV. 

L,UKK£,t 
1.2 

_HJNLr  ?  AC  1  UK.         -  T  /y.   L.ON 
i-3           i-4 

TINUED 
1-5 

0 

0 

0 

0 

O 

o 

45 

1.0081 

11.519 

I.OII5 

13.692 

1.0158 

16.034 

1.  02  1  2 

18.568 

1.0279 

21.262 

46 

I.OOII 

"•543 

I.003I 

13.726 

i.  0060 

16.092 

I.OO99 

18.639 

1.0148 

21-355 

47 

0.99409 

"•555 

0-99475 

I3.748 

0-99623 

16.128 

0.99850 

18.689 

I.OOlS 

21.426 

48 

0.98709 

"•553 

0.98650 

13-753 

0.98654 

16.142 

0.98729 

18.717 

0.98893 

21-474 

49 

0.98018 

0.97825 

I3.74I 

0.97685 

16.137 

0.97607 

18.723 

0.97613 

2  1  .496 

50 

0.97327 

11.506 

0.97008 

13.712 

0.96722 

16.113 

0.96493 

18.707 

0.96340 

21.492 

51 

0.96636 

11.461 

0.96192 

13.666 

0.95770 

16.068 

0-95393 

18.668 

0.95080 

21.462 

52 

0.95964 

11.401 

0.95383 

13.603 

0.94822 

16.004 

0.94300 

18.605 

0.93833 

21.405 

53 

0.95291 

11.327 

0-94583 

I3.523 

0.93892 

I5-9I9 

O.9322I 

18.519 

0.92593 

21.321 

54 

0.94627 

11.239 

0.93800 

I3.42S 

0.92969 

15.814 

0.92157 

18.409 

0.91373 

2I.2IO 

55 

0.93963 

11.136 

0.93017 

I3.3IO 

0.92054 

15.688 

O.9IIOO 

18.275 

0.90167 

2I.O72 

56 

0.93318 

11.019 

0.92250 

13.178 

0.91154 

I5.542 

0.90064 

18.117 

0.88980 

20.907 

57 

0.92682 

10.888 

0.91492 

13.028 

0.90269 

I5-376 

0.89043 

17.936 

0.87813 

20.713 

58 

0.92054 

10.742 

0.90750 

12.860 

0.89400 

15.188 

0.88036 

17-730 

0.86667 

2O.49I 

59 

0.91436 

10.581 

0.90017 

12.675 

0.88546 

14.980 

0.87050 

17.500 

0.85540 

2O.240 

6o 

0.90836 

10.406 

0.89300 

12.474 

0.87715 

14-752- 

0.86086 

17-246 

0-84433 

19.964 

61 

0.90244 

10.218 

0.88600 

12.255 

0.86892 

14.502 

0.85136 

16.967 

0-83353 

I9-657 

62 

0.89666 

10.016 

0.87918 

12.019 

0.86092 

14.232 

0.84214 

!6.663 

0.82300 

19.321 

63 

0.89103 

9.800 

0.87250 

11.767 

0.85315 

13-942 

0.83314 

16.335 

0.81273 

18.956 

64 

0.88554 

9-570 

0.86600 

11.497 

0.84554 

13.632 

0.82436 

15-983 

0.80267 

18.562 

65 

0.88019 

9-327 

0.85967 

II.  211 

0.83808 

I3.30I 

0.81579 

15.607 

0.79293 

18.140 

66 

0.87501 

9.071 

0.85350 

10.909 

0.83094 

12.950 

0.80750 

15.207 

0.78347 

17.690 

67 

0.86998 

8.801 

0.84758 

10.594 

0.82400 

12.580 

0.79950 

14-783 

0-77433 

17.211 

68 

0.86514 

8.519 

0.84183 

10.257 

0.81731 

12.191 

0.79171 

14-335 

0.86547 

16.704 

69 

0.86045 

8.225 

0.83625 

9.907 

0.81077 

11.783 

0.78421 

13-865 

0.75693 

16.169 

70 

0.85596 

7.918 

0.83093 

9-543 

0.80454 

"•357 

0.77707 

13-373 

0.74873 

15.607 

71 

0.85165 

7.600 

0.82583 

9.164 

0.79862 

10.912 

O.77O2I 

12.858 

0.74087 

I5.0I8 

72 

0.84754 

7.271 

0.82095 

8-771 

0.79292 

10.450 

0.76364 

12.322 

0-73333 

14.402 

73 

0.84362 

6.931 

0.81631 

8.365 

0.78746 

9.971 

0.75736 

11.765 

0.72620 

I3.76I 

74 

0.83989 

6.580 

0.81190 

7.946 

0.78231 

9-476 

11.188 

0.71940 

13.096 

75 

0.83637 

6.  2  2O 

0.80773 

7.514 

0.77746 

8.966 

0-74579 

10.591 

0.71293 

12.406 

76 

0.83306 

5.850 

0.80380 

7.069 

0.77287 

8.440 

0.74050 

9-975 

0.70687 

11.693 

77 

0.82996 

5-471 

0.80013 

6.614 

0.76858 

7.900 

0-73557 

9-342 

0.70120 

10.958 

78 

0.82707 

5.084 

0.79671 

6.148 

0.76459 

7-346 

0.73093 

8.692 

0-69593 

10.201 

79 

0.82441 

4.689 

0-79355 

5.672 

0.76090 

6.780 

0.72664 

8.026 

0.69107 

9-425 

80 

0.82196 

4.286 

0.79065 

5-187 

0-75751 

6.202 

0.72279 

7-346 

0.68660 

8.630 

81 

0.81975 

3.877 

0.78802 

4-693 

0-75444 

5.614 

0.71921 

6.651 

0.68253 

7.8l8 

82 

0.81775 

3.462 

0.78566 

4.191 

0-75165 

5.016 

0.71600 

5-944 

0.67887 

6.991 

83 

0.81599 

3.041 

0.78358 

3-683 

0.74923 

4.408 

0.71319 

5.227 

0.67567 

6.149 

84 

0.81445 

2.616 

0.78175 

3.169 

0-747" 

3-793 

0.71073 

4-499 

0.67287 

5-294 

85 

0.81315 

2.186 

0.78022 

2.649 

0-74531 

3-i7i 

0.70864 

3-762 

0.67047 

4.429 

86 

0.81209 

1-753 

0.77895 

2.124 

0.74383 

2-544 

0.70694 

3.018 

0.66847 

3-554 

87 

0.81125 

1.317 

0-77797 

1.596 

0.74268 

1.912 

0.70561 

2.269 

0.66693 

2.673 

88 

0.81066 

0.879 

0.77727 

1.065 

0.74185 

1.277 

0.70466 

1.515 

0.66587 

1-785 

89 

0.81031 

0.440 

0.77684 

0-533 

0.74136 

0.639 

0.70409 

0.758 

0.66521 

0.893 

90 

0.81019 

o.ooo 

0.77670 

o.ooo 

0.74120 

o.ooo 

0.70389 

o.ooo 

0.66499 

o.ooo 

E: 

camole. 

sinh  (1.5 

765°) 

n  ^rnn?  / 

-  o  *7n/?n-5 

/TR°  <•>«':•: 

/65° 


[22] 


1ABLK  IV.   l^UJK.JK.Jl,V^lliN»j 

1.6                1.7 

r-rt^lUJ*..     -    m  '  IJk   CONTINUED 
1.8              1.9              2.O 

o 

o 

b 

0 

0 

o 

45 

1-0359 

24.117 

1.0456 

27.I33 

1.0572 

30.292 

1.0707 

33.590 

1.0863 

37.016 

46 

1.  02  1  1 

24.241 

1.0290 

27.288 

1.0385 

30.486 

1.0499 

33.829 

1-0633 

37.304 

47 

I.O064 

24-339 

1.0124 

27.418 

1.0199 

30.651 

1.0292 

34.037 

1.0405 

37.560 

48 

0.99175 

24.409 

0.99582 

27.5I5 

1.0015 

30.784 

1.0087 

34-2io 

1.0178 

37.778 

49 

0.97719 

24451 

0.97947 

27.582 

0.98317 

30.883 

0.98837 

34-346 

0-99535 

37.958 

50 

0.96275 

24-465 

0.96323 

27.618 

0.96500 

30.946 

0.96821 

34-444 

0.97310 

38.098 

Si 

0.94844 

24-449 

0.94718 

27.620 

0.94700 

30.974 

0.94821 

34-503 

0.95110 

38.196 

52 

0.93431 

24.402 

0.93123 

27.590 

0.92922 

30.966 

0.92858 

34-524 

0.92940 

38.253 

53 

0.92031 

24-325 

0.91547 

27.527 

0.91167 

30.921 

0.90911 

34.505 

0.90800 

38.267 

54 

0.90650 

24.218 

0.89994 

27.429 

0.89433 

30-839 

0.88995 

34-444 

0.88685 

38.236 

55 

0.89281 

24.079 

0.88465 

27.296 

0.87728 

30.718 

0.87106 

34-342 

0.86600 

38.160 

56 

0.87938 

23.910 

0.86953 

27.128 

0.86044 

30.558 

0.85242 

34-197 

0.84555 

38.039 

57 

0.86619 

23.709 

0.85471 

26.924 

0.84389 

30.358 

0.83411 

34.008 

0.82545 

37.869 

58 

0.85319 

23-475 

0.84012 

26.683 

0.82767 

30.116 

0.81611 

33-773 

0.80570 

37-650 

59 

0.84044 

23.209 

0.82582 

26.406 

0.81172 

29-834 

0.79847 

33-493 

0.78630 

37.382 

60 

0.82794 

22.910 

0.81176 

26.091 

0.79611 

29.510 

0.78121 

33-I67 

0.76735 

37.062 

61 

0.81569 

22.578 

0.79806 

25.738 

0.78083 

29.142 

0.76437 

32.792 

0.74880 

36.688 

62 

0.80375 

22.212 

0.78464 

25-347 

0.76594 

28.730 

0-74795 

32-368 

0.73070 

36.261 

63 

0.79213 

2I.8I3 

0.77159 

24.917 

0.75139 

28.274 

0.73179 

31-803 

0.71300 

35.778 

64 

0.78075 

21.380 

0.75888 

24.447 

0.73722 

27.774 

0.71611 

3I-368 

0-69575 

35.238 

65 

0.76975 

20.913 

0.74647 

23.938 

0.72344 

27.228 

0.70084 

30.792 

0.67900 

34.639 

66 

0.75906 

20.413 

0.73447 

23.390 

0.71006 

26.636 

0.68605 

30.162 

0.66275 

33.980 

67 

0.74869 

19.879 

0.72282 

22.802 

0.69711 

25-997 

0.67168 

29.478 

0.64700 

33-261 

68 

0.73869 

19.311 

0.71159 

22.175 

0.68455 

25-3" 

0.65779 

28.741 

0.63175 

32.479 

69 

0.72900 

I8.7IO 

0.70076 

21.508 

0.67244 

24.580 

0.64421 

27.949 

0.61700 

31-634 

70 

0.71975 

18.076 

0.69035 

20.801 

0.66083 

23.802 

0.63153 

27.102 

0.60280 

30.725 

0.71081 

17.410 

0.68035 

20.055 

0.64967 

22.976 

0.61916 

26.199 

0.58915 

29-750 

72 

O.7023I 

16.712 

0.67076 

19.271 

0.63900 

22.104 

0.60731 

25.241 

0.57605 

28.709 

73 

0.69419 

15.982 

0.66165 

18.448 

0.62878 

21.187 

0.59600 

24.227 

0-56355 

27.602 

74 

0.68650 

15.221 

0.65300 

17.588 

0.61911 

20.224 

0.58526 

23-158 

o-55l6S 

26.428 

75 

0.67919 

I4.43I 

0.64476 

16.692 

0.60994 

19.215 

0.57500 

22.034 

0-54035 

25-187 

76 

0.67231 

I3.6I3 

0.63700 

15.760 

0.60128 

18.162 

0.56542 

20.856 

0.52970 

23.880 

77 

0.66588 

12.767 

0.62976 

14.794 

0.59317 

17.068 

0.55637 

19.625 

0.51970 

22.507 

78 

0.65988 

11.894 

0.62300 

13-795 

0.58561 

15-933 

0-54795 

18-343 

0-51035 

21.070 

79 

0.65431 

10.997 

0.61676 

12.765 

0.57861 

I4.758 

0.54011 

17.012 

0.50165 

19-571 

80 

0.64925 

IO.O76 

0.61106 

11.705 

0.57217 

13-545 

0.53295 

I5-633 

0.49364 

18.01  1 

81 

0.64463 

9-J34 

0.60582 

10.618 

0.56633 

12.298 

0.52642 

14.209 

0.48636 

16.391 

82 

0.64050 

8.171 

0.60112 

9-505 

0.56106 

11.019 

0.52052 

12-744 

0.47979 

14.722 

83 

0.63681 

7.191 

o.597oo 

8.370 

0.55639 

9.711 

0-51530 

11.240 

0.47396 

13.000 

84 

0.63363 

6.194 

0-59341 

7.214 

0.55234 

8-374 

0.51075 

9.702 

0.46888 

11.233 

85 

0.63094 

5.184 

0.59035 

6.039 

0.54890 

7-oi5 

0.50689 

8-133 

0.46455 

9.426 

86 

0.62875 

4.161 

0.58782 

4.850 

0.54607 

5-636 

0.50372 

6-539 

0.46101 

7.584 

87 

O.627OO 

3.129 

0.58586 

3-649 

0.54387 

4.241 

0.50124 

4-923 

0.45823 

5-713 

88 

0.62575 

2.090 

0.58445 

2.438 

0.54229 

2-834 

0.49947 

3.291 

0.45624 

3-821 

89 

O.6250O 

1.046 

0.58361 

1.230 

0.54135 

1.419 

0.49841 

1.648 

0.45505 

1.914 

90 

0.62473 

o.ooo 

0.58333 

o.ooo 

0.54103 

o.ooo 

0.49805 

o.ooo 

0.45465 

o.ooo 

Example. 

sinh  (i 

•8/77°) 

O  CO?  I  7  , 

^I7°.o68 

=  O.CQ?!?  /I7°.4' 

.05'. 

1.8/77° 

<~'oyoi  /  i 

[23] 


sinh  6 


1ABLE  IV. 
2.1 

UUiVKJl,^  JL  1IN  (j 
2.2 

rA^iUK..  -     -  r  77.  CON 

2.3          2.4 

TINUED 
2-5 

o 

o 

0 

o 

o 

O 

45 

1.1043 

40.558 

I.I248 

44.205 

1.1480 

47.946 

1.1740 

5L769 

1.2032 

55-661 

46 

1.0789 

40.905 

1.0970 

44.613 

1.1177 

48.419 

1.1411 

52.312 

1.1674 

56.278 

47 

1.0538 

41.213 

1.0695 

44.981 

1.0877 

48.851 

1.1085 

52.813 

1.1320 

56.852 

48 

1.0289 

41.482 

1.0423 

45-307 

1.0580 

49.241 

1.0762 

53.27I 

1.0972 

57.383 

49 

1.0043 

41.711 

I.OI53 

45-592 

1.0286 

49.588 

1.0444 

53-685 

1.0627 

57.869 

50 

0.97986 

41.898 

0.98864 

45-834 

0.99965 

49.890 

1.0130 

54-053 

J.0288 

58.309 

Si 

0.95576 

42.042 

0.96232 

46.031 

0.97104 

50.146 

0.98200 

54-375 

0.99540 

58.701 

52 

0.93195 

42.142 

0.93636 

46.181 

0.94283 

50.354 

0.95150 

54-648 

0.96256 

59.044 

53 

0.90848 

42.197 

0.91077 

46.284 

0.91504 

50.5H 

0.92154 

54-871 

0.93028 

59-337 

54 

0.88533 

42.205 

0.88559 

46.338 

0.88774 

50.623 

0.89208 

55-042 

0.89860 

59-578 

55 

0.86257 

42.165 

0.86082 

46.342 

0.86091 

50.680 

0.86308 

55.i6o 

0.86748 

59-765 

56 

0.84019 

42.076 

0.83646 

46.294 

0-83457 

50.683 

0.83467 

55-223 

0.83700 

59.896 

57 

0.81819 

41.935 

0.81255 

46.193 

0.80869 

50.629 

0.80683 

55-228 

0.80712 

59-970 

58 

0.79662 

41.742 

0.78909 

46.035 

0.78331 

50.518 

0-77954 

55-174 

0.77788 

59.984 

59 

0-77543 

41.494 

0.76609 

45.820 

0.75848 

50.347 

0.75283 

55-059 

0.74932 

59-936 

60 

0.75471 

41.191 

0-74359 

45-546 

o.734i7 

50.114 

0.72671 

54.880 

0.72140 

59-823 

61 

0-73447 

40.830 

0.72159 

45-2io 

0.71043 

49.816 

0.70121 

54.634 

0.69412 

59.642 

62 

0.71467 

40.410 

O.7OOO9 

44.811 

0.68725 

49-451 

0.67633 

54-3I9 

0.66752 

59-391 

63 

0.69538 

39.929 

0.67914 

44-345 

0.66465 

49.016 

0.65204 

53-931 

0.64160 

59.067 

64 

0.67657 

39-386 

0.65873 

43.811 

0.64261 

48.509 

0.62838 

53-467 

0.61636 

58.665 

65 

0.65829 

38.777 

0.63886 

43.207 

0.62113 

47.926 

0.60533 

52.924 

0.59176 

58.182 

66 

0.64053 

38.101 

0.61955 

42.529 

0.60026 

47.264 

0.58296 

52-297 

0.56784 

57.613 

67 

0.62329 

37-357 

O.6OO82 

41-775 

0.58004 

46.519 

0.56121 

51-583 

0.54464 

56.954 

68 

0.60657 

36-542 

0.58268 

40.943 

0.56043 

45-688 

0.54013 

50.777 

0.52208 

56.198 

69 

0.59043 

35-655 

0.56514 

40.029 

0.54143 

44.767 

0.51971 

49.874 

0.50024 

55-341 

70 

0.57487 

34-694 

0.54818 

39.030 

0.52309 

43-752 

0.49996 

48.868 

0.47908 

54-376 

71 

0.55990 

33-656 

0.53186 

37-944 

0-50539 

42.638 

0.48088 

47-754 

0.45864 

72 

0-54557 

32.541 

0.51618 

36.769 

0.48835 

41.422 

0.46246 

46.526 

0.43888 

52.092 

73 

0.53181 

3L348 

0.50118 

35-501 

0.47204 

40.100 

0.44479 

45-177 

0.41988 

50.755 

74 

0.51871 

30.074 

0.48682 

34.I38 

0.45644 

38-667 

0.42783 

43-701 

0.40160 

49-277 

75 

0.50629 

28.719 

'0.47323 

32-678 

0.44152 

37.118 

0.41165 

42.091 

0.38408 

47.648 

76 

0-49453 

27.283 

0.46027 

31.119 

0.42736 

35-449 

0.39621 

40.341 

0.36732 

45-857 

77 

0.48347 

25.766 

0.44810 

29-459 

0.41397 

33-659 

0.38157 

38.443 

0.35137 

43-891 

78 

0.47311 

24.168 

0.43666 

27.699 

0.40135 

31-744 

0.36775 

36.390 

0.33625 

41.740 

79 

0.46349 

22.491 

0.42601 

25-839 

0.38963 

29.701 

0.35480 

34-I78 

0.32201 

39-392 

80 

0.45461 

20.736 

0.41617 

23.879 

0.37873 

27.531 

0.34273 

31-803 

0.30869 

36.836 

81 

0.44650 

18.907 

0.40716 

21.822 

0.36873 

25-233 

0.33162 

29.261 

0.29633 

34-063 

82 

0.43919 

17.008 

0.39902 

19.671 

0-35965 

22.810 

0.32149 

26.550 

0.28500 

31-065 

83 

0.43268 

15-043 

0.39176 

I7-432 

0.35153 

20.267 

0.31239 

23.676 

0.27477 

27.841 

84 

0.42699 

13-015 

0.38540 

15.111 

0.34441 

17.611 

0.30438 

20.643 

0.26569 

24.390 

85 

0.42215 

10.933 

0.37998 

12.715 

0.33831 

14.851 

0.29748 

17.462 

0.25784 

20.723 

86 

0.41818 

8.806 

0-37551 

10.254 

0.33328 

I2.OOO 

0.29178 

14.147 

0.25131 

16.855 

87 

0.41507 

6.639 

O.372O2 

7.740 

0.32933 

Q.07I 

0.28728 

10.717 

0.24614 

12.811 

88 

0.41284 

4-443 

0.36951 

5-183 

0.32650 

6.082 

0.28405 

7.197 

0.24241 

8.624 

89 

0.41150 

2.226 

0.36800 

2-599 

0.32479 

3-05I 

0.28210 

3-615 

0.24014 

4.338 

90 

0.41105 

o.ooo 

0.36750 

o.ooo 

0.32422 

O.OOO 

0.28144 

o.ooo 

0.23939 

0.000 

Example. 

sinh  (2. 

3/84!) 

=  n  7/1/1/1  T  , 

f  T  *7^  fkT  T 

/I7°  36'. 

ff 

2.3/84°     -«™->-i  -" 

' 

/  *  /  *ov 

[24] 


1ABLE  J.V. 
2.6 

^UJS.K.H,^  1  1IN  Vj 

2-7 

r  A^  J.  U.K.. 

0 

2.8 

-  r  77.   CONTINUED 
2.9             3.0 

0 

o 

e 

o 

o 

o 

45 

1.2354   59-6I3 

1.2710  63.614 

I.3I02   67.653 

1-3530   7L72I 

1-3995   75.8I4 

46 

1.1967   60.307 

1.2293   64.386 

1.2652   68.506 

1.3046   72.658 

1-3475   76-832 

47 

1.1586   60.957 

1.1881   65.116 

I.22O9   69.317 

1.2570   73-551 

1.2965   77.808 

48 

I.I209   61.564 

1.1476  65.801 

I.I774   70.084 

1.2103   74-400 

1.2465   78.741 

49 

1.0838   62.126 

1.1077  66.442 

I.I345   70.806 

1.1644   75.205 

1.1974   79-630 

50 

1.0473   62.641 

1.0685  67.037 

1.0925   71.483 

1.1195   75.965 

1.1494   80.474 

Si 

I.OII3   63.109 

1.0300  67.585 

1.0513   72.113 

1-0755   76.679 

1.1025   81.272 

52 

0.97608  63.528 

0.99219  68.084 

I.OIO9   72.694 

1.0324   77-345 

1.0567   82.024 

53 

0.94145  63.897 

0.95514  68.532 

0.97143  73-226 

0.99038  77-963 

I.OI2O    82.729 

54 

0.90750  64.214 

0.91881  68.929 

0.93279  73-708 

0.94928  78.532 

0.96837   83.386 

55 

0.87419  64.476 

0.88333  69.274 

0.89500  74-138 

0.90917  79.050 

0.92590   83.994 

56 

0.84162  64.684 

0.84863  69.563 

0.85811  74.515 

0.87007  79-5I7 

0.88453   84.553 

57 

0.80973  64.834 

0.81467  69.797 

0.822II  74.837 

0.83197  79-932 

0.84433   85.062 

58 

0.77854  64.925 

0.78152  69.973 

0.78700  75.103 

0.79487  80.293 

0.80523   85.510 

59 

0.74808  64.954 

0.74919  70.088 

0.75275  75-3H 

0.75876  80.598 

0.76727   85.924 

Oo 

0.71835  64.918 

0.71767  70.140 

0.71943  75-459 

0.72366  80.847 

0.73040   86.276 

61 

0.68935  64.816 

0.68692  70.128 

0.68700  75.544 

0.68955  81.037 

0.69463   86.574 

62 

0.66104  64.644 

0.65700  70.047 

0.65543  75-565 

0.65641  81.167 

0.65993   86.8l7 

63 

0.63346  64.399 

0.62786  69.895 

0.62475  75.5I9 

0.62424  81.235 

0.62630   87.004 

64 

0.60662  64.077 

0.59952  69.669 

0-59493  75-403 

0.59303  81-239 

0-59373   87.133 

65 

0.58054  63.674 

0.57193  69.364 

0.56596  75.214 

0.56272  81.176 

0.56217   87.204 

66 

0.55519  63.185 

0.54514  68.978 

0.53785  74-948 

0-53334  81.044 

0.53160   87.214 

67 

0.53054  62.606 

0.51911  68.504 

0.51054  74-599 

0.50486  80.838 

O.5O2O3   87.161 

68 

0.50658  61.930 

0.49386  67.935 

0.48404  74.164 

0.47724  80.555 

0.47340   87.043 

69 

0.48338  61.151 

0.46937  67.266 

0.45832  73.634 

0.45045  80.190 

0.44567   86.858 

70 

0.46089  60.262 

0.44559  66.489 

0-43343  73-ooS 

0.42448  79.738 

0.41883   86.602 

0.43911  59.254 

0.42256  65.596 

0.40928  72.267 

0-39934  79-191 

0.39283   86.271 

72 

0.41808  58.117 

0.40026  64.576 

0.38586  71.411 

0-37497  78.541 

0.36767   85.860 

73 

0-39773  56-841 

0.37872  63.415 

0.36321  70.424 

0.35135  77.780 

0.34327   85.363 

74 

0-37814  55-4I4 

0.35789  62.101 

0.34125  69.292 

0.32845  76.894 

0.31964   84.772 

75 

0.35930  53-822 

0.33780  60.617 

0.32002  67.996 

0.30627  75.867 

0.29674   84.077 

76 

0-34123  52.049 

0.31846  58.943 

0.29951  66.516 

0.28478  74.680 

0.27452   83.265 

77 

0.32395  50-079 

0.29987  57.058 

0.27971  64.827 

0.26397  73.309 

0.25296   82.319 

78 

0.30749  47-893 

0.28206  54.935 

0.26064  62.897 

0.24379  71.721 

0.23202  81.216 

79 

0.29189  45.471 

0.26507  52.542 

0.24231  60.686 

0.22432  69.876 

0.21168  79.925 

80 

0.27718  42-793 

0.24894  49-847 

0.22476  58.145 

0.20551  67.721 

0.19191  78.404 

81 

0.26346  39.837 

0.23374  46.812 

0.20804  55-217 

0.18738  65.183 

0.17270  76.592 

82 

0.25078  36.583 

0.21954  43-396 

0.19222  51.833 

0.16998  62.169 

0.15404  74-404 

83 

0.23922  33.016 

0.20644  39.554 

0.17741  47.908 

0.15343  58.553 

0.13596  71-709 

84 

0.22887  29.126 

0.19458  35.252 

0.16375  43-351 

0.13774  54.i68 

0.11849  68.313 

85 

0.21986  24.910 

0.18409  30.458 

0.15142  38.066 

0.12323  48.795 

0.10174  63.903 

86 

0.21227  20.382 

0.17515  25.164 

0.14069  31.968 

0.11016  42.168 

0.085917  57-974 

87 

0.20623  15.571 

0.16794  19.390 

0.13184  25.015 

0.098934  34-005 

0.071443  49-701 

88 

0.20184  10.524 

0.16264  13.194 

0.12520  17.244 

0.090148  24.110 

0.059103  37.843 

89 

0.19917  5.307 

0.15938  6.682 

0.12105  8.809 

0.084469  12.576 

0.050307  21.091 

90 

0.19827  o.ooo 

0.15829  o.ooo 

0.11964  o.ooo 

0.082845  o.ooo 

0.047040  o.ooo 

Example. 

sinh  (3.0  /86°) 

0.085917  /57°-974 

J'.26*. 

3-Q  /86° 

[25] 


TABLE 

V.  CORRECTING  FACTOF 

tanh 

ti. 

0 

O.I 

0.2 

0.3 

0.4 

o. 

5 

0 

45 
46 
47 
48 
49 

I.OOOOO 
I.OOOIO 
I.OO02I 

1.00033 
1.00044 

o 

o.i  88 

0.188 
0.188 
0.188 
0.187 

0.99985 
I.O003O 
I.OO075 
I.OOI2O 
I.OOI7O 

0 

0.765 
0.764 
0.763 
0.76l 
0.758 

0.99937 

1.00040 

1.00143 

1.00250 

1.00357 

0 

I.7I8 
I.7I9 
1.720 
I.7I7 
I.7IO 

0.99803 
0.99988 
1.00173 
1.00358 
1.00543 

o 

3.046 
3.055 
3.058 
3.058 
3.053 

0.99514 

0.99804 
1.00094 
1.00384 
1  .00680 

o 

4.750 
4.771 
4.781 
4.787 
4.787 

So 

52 

53 
54 

1.00056 
1.00068 
1.00080 
1.00092 
1.00103 

0.187 
0.187 
0.186 
0.184 
0.182 

I.OO2I5 
I.OO265 
I.O03IO 
1-00355 
I.OO4O5 

0-755 
0.749 
0.744 
0.738 
0.732 

1.00460 
1.00563 
1.00670 

1.00773 
1.00877 

1-703 
1.694 
1.685 
1.672 
1.656 

1.00733 
1.00925 
1.01115 
1.01303 
1.01490 

3.045 
3.031 
3.014 
2.995 
2.969 

1  .00980 
1.01278 

1.01578 
1.01878 

1.02178 

4.780 
4.767 

4-747 
4.724 
4.692 

55 
56 
57 
58 
59 

1.00114 
1.00125 

1.00135 

1.00146 
1.00156 

0.180 
0.177 
o.i75 
0.172 
0.170 

1.00450 
1.00495 
1-00535 
1-00575 
I.OO62O 

0.724 

0.715 
0.705 
0.694 
0.682 

1.00980 
1.01083 
1.01183 
1.01283 
1.01380 

1.638 

1.618 
1.596 

1-574 
1.548 

1.01675 
1.01865 
1.02050 
1.02230 
1.02410 

2.942 

2.911 

2.876 
2.836 
2.793 

1.02478 
1.02778 
1.03076 

1-03374 
1.03670 

4-656 
4.612 
4.561 
4-506 
4-444 

6o 
61 
62 
63 
64 

1.0017 
1.0018 
1.0019 

I.OO20 
I.OO2O 

0.167 
0.163 
0.159 
0.155 
0.151 

1.0067 
1.0070 
1.0074 
1.0078 
I.OO82 

0.668 
0.655 
0.640 
0.625 
0.609 

1.0148 

1.0158 

1.0167 
1.0176 

1.0185 

1.521 
1.490 
1.458 
1-425 
1.390 

1.0259 
1.0278 
1.0294 
1.0311 
1.0328 

2.745 
2.695 

2.641 

2.583 
2.521 

1.0397 
1.0425 
1.0454 

1.0482 
1.0510 

4-375 
4-300 
4.220 
4-133 
4.041 

65 
66 

67 
68 
69 

I.OO2I 
I.OO22 
1.0023 
1.0024 
I.OO25 

0.147 
0.143 
0.138 
0.134 
0.129 

I.  OO86 
I.OOgO 
1.0093 
I.OO97 
I.OIOO 

0.592 
0-575 
0-557 
0-539 
0.520 

1.0193 

I.02O2 
I.O2IO 

1.0218 

I.O226 

1-353 
I.3I4 
1-273 
1.231 
1.187 

1.0344 
1.0360 
1.0376 
1.0391 
1.0406 

2.456 
2.388 
2.315 

2.240 
2.163 

1-0538 
1.0564 
1.0591 

1.0617 
1.0642 

3-942 
3-836 
3-724 
3-609 
3-489 

70 

72 
73 
74 

I.O026 
1.0026 
1.0027 
I.OO28 
I.OO28 

0.123 
0.118 

O.II2 
O.I07 
O.IO2 

1.0103 

1.0106 
1.0109 

I.OII2 

1.0114 

0-499 
0.478 
0.456 
0.434 
0.411 

1.0234 
I.O24I 
1.0247 
1.0254 
I.O260 

1.141 
1.094 
1.045 
0-995 
0.944 

1.0420 
1.0434 
1.0447 
1.0460 
1.0472 

2.082 
1.998 

1.911 
1.821 
1.728 

1.  0666 
1.0689 
1.0712 

1-0734 

1-0755 

3-362 
3.229 
3.091 
2.948 
2.800 

75 
76 
77 
78 
79 

I.OO29 
I.OO29 
1.0030 
I.OO3O 
I.003I 

O.O96 
O.O9O 
0.084 
0.078 
O.072 

1.0117 

1.  01  2O 
I.OI22 
I.OI23 
I.OI25 

0.388 
0.365 
0.341 
0.316 
0.291 

1.0266 
I.O272 
1.0277 
I.O282 
I.O287 

0.892 
0.838 
0.783 
0.727 
0.671 

1.0483 
1.0494 
1.0504 

1-0513 
1.0522 

1.632 
1-534 
1-434 
1-332 
1.229 

1.0774 
1.0793 
1.081  1 
1.0828 
1.0843 

2.649 
2-493 
2-333 
2.169 

2.OOI 

80 
81 
82 
83 
84 

I.003I 
1.0032 
1.0032 
1.0032 
1.0033 

O.066 
O.O6O 
0.054 
0.047 
O.O4O 

I.OI27 
I.OI29 
I.OI30 
I.OI32 
I-OI33 

0.266 
0.241 
0.215 
o.i  88 
0.162 

I.O29I 
1.0295 
1.0298 
I.030I 
1.0304 

0.614 
0-555 
0-494 
0-433 
0.372 

1-0530 
1-0538 
1.0544 
1-0550 
1-0555 

1.124 
1.017 
0.908 
0.797 
0.685 

1.0857 
1.0870 
i.  088  1 
1.0892 
1.0901 

.830 
.656 
.480 
.301 
.119 

85 
86 

87 
88 
89 

1-0033 
1.0033 
1-0033 
1.0033 
1-0033 

0.033 
O.O26 
0.019 
O.OI2 
O.OO6 

I.OI34 
I.OI34 
I-OI35 
I-OI35 
I-OI35 

0.136 
0.109 
0.082 
0.054 
0.027 

1.0307 
1.0309 
I.03IO 
I.03II 
I.03II 

0.311 
0.249 
0.187 
0.125 
0.063 

1.0560 
1-0563 
1.0566 
1.0568 
1.0570 

0-573 
0.460 
0.346 
0.232 
0.117 

i  .0908 
1.0914 
1.0919 
1.0923 
1.0926 

0-935 
0-749 
0.562 

0-374 
0.187 

90 

1.0033 

O.OOO 

I.OI36 

O.OOO 

I-03II 

O.OOO 

1.0570 

O.OOO 

1.0926 

0.000 

Note. 
Example. 

Negative  quantities  are  ii 
tanh  (0.4  /74°) 

i  heavy 
\i°.728 

type. 

/  0 

=  1.0472 

-  1.0472  \i-.43 

41*. 

[26] 


TABLE  V.     CORRECTING 


0.6 


0.7 


tanhfl 
0 


•  /y.    CONTINUED 


0.8 


0.9 


45 
46 

47 

0.99010 
0.99417 
0.99830 

6.817 

6.854 
6.881 

0.98189 
0.98727 
0.99279 

9.214 
9.281 
9-337 

0.96971 
0.97646 
0-98339 

11.902 
I2.O2O 
12.122 

0.95284 
0.96089 
0.96921 

14.839 
15.020 
15-185 

0.93077 
0.93999 
0.94950 

17.956 
18.216 
18.461 

48 

1.00248 

6.901 

0.99840 

9-383 

0.99050 

12.210 

0.97779 

15-334 

0.95938 

18.692 

49 

1.00672 

6.912 

I 

00410 

9.419 

0.99780 

12.285 

0.98665 

15.469 

0.96966 

18.908 

50 

I.OIIOO 

6.915 

I 

00991 

9-443 

1.00526 

12.347 

0.99578 

15-588 

0.98032 

19.108 

1-01533 

6.908 

I 

01581 

9-455 

1.01289 

12.395 

1.00518 

15.690 

0.99136 

19.290 

52 

1.01970 

6.893 

I 

02181 

9-457 

1.02069 

12.427 

1.01486 

15-777 

1.00282 

19-455 

53 

1.02412 

6.870 

I 

02789 

9-445 

1.02864 

12.444 

1.02479 

15-845 

1.01469 

19.600 

54 

1.02855 

6.838 

i 

03403 

9-423 

1.03675 

12.446 

1.03500 

15.895 

1.02697 

19.726 

55 

1.03300 

6.797 

I 

04024 

9-388 

1.04501 

12.432 

1.04549 

15-925 

1.03970 

19.828 

56 

1.03747 

6.746 

i 

04653 

9-338 

1-05343 

I2.4OI 

1.05626 

15-934 

1.05287 

19.909 

57 

1.04195 

6.685 

1.05286 

9-275 

1.06196 

12.353 

1.06729 

15-922 

1.06648 

19.965 

58 

1.04642 

6.616 

I 

05921 

9.200 

1.07061 

12.288 

1.07854 

15-889 

1.08054 

19.996 

59 

1.05089 

6-537 

I 

06561 

9.112 

1.07939 

12.203 

1.09007 

15.833 

1.09506 

19.998 

60 

1-05537 

6.448 

i 

07210 

9.010 

1.08829 

I2.IOO 

1.10188 

15-753 

1.11009 

19-973 

61 

1.05980 

6.350 

i 

07856 

8.893 

1.09726 

11.978 

1.11390 

15-647 

1-12555 

19.918 

62 

1.06420 

6.242 

1.08500 

8.762 

1.10630 

11.835 

1.12614 

15-514 

1.14144 

19-832 

63 

1.06858 

6.124 

I 

09H3 

8.617 

1.11540 

11.673 

1.13856 

15-355 

I-I5777 

19.711 

64 

1.07293 

5-998 

I 

09783 

8.458 

1.12456 

11.491 

1.15116 

15-169 

I.I7454 

19.555 

65 

1.07722 

5.862 

I 

10420 

8.285 

I.I3373 

11.288 

1.16392 

14.956 

1.19173 

19-362 

66 

1.08143 

5-7i6 

I 

"053 

8.098 

1.14291 

11.064 

1.17684 

14.711 

1.20935 

19.131 

67 

1.08560 

5-561 

I 

11683 

7.895 

1.15208 

IO.82O 

1.18989 

14.438 

1.22737 

18.860 

68 

1.08967 

5-397 

i 

12299 

7.678 

1.  16118 

10.552 

1.20298 

14-132 

1.24569 

18.545 

69 

1.09363 

5-223 

I 

12904 

7-447 

1.17020 

10.263 

1.21608 

13-796 

1.26429 

18.188 

70 

1.09750 

5.041 

I 

13500 

7.202 

I.I79I5 

9-953 

1.22919 

13-428 

1.28316 

17-785 

1.10125 

4-850 

I 

14083 

6-943 

1.18796 

9.621 

1.24226 

13.026 

1.30221 

17-334 

72 

1.10490 

4-650 

I 

14651 

6.671 

1.19659 

9.268 

1-25524 

12.592 

1.32140 

16.836 

73 

1.10842 

4.441 

i 

15201 

6.386 

1.20503 

8.894 

1.26807 

12.126 

1.34063 

16.290 

74 

1.11180 

4-225 

I 

15734 

6.087 

1.21325 

8-499 

1.28067 

11.628 

1.35986 

I5-693 

75 

1.11503 

4.002 

1.16244 

5-775 

I.22I2I 

8.083 

1.29300 

11.097 

1.37894 

15-043 

76 

1.  11810 

3-772 

1-16733 

5-450 

1.22889 

7.648 

1.30500 

10.532 

1-39775 

14.342 

77 

I.I2I02 

3-534 

I 

17197 

5-II4 

1.23623 

7.192 

1-31659 

9-936 

1.41620 

13-591 

I-I2375 

3.289 

1.17636 

4.767 

I.243I9 

6.718 

1.32772 

9-309 

1.43412 

12.788 

79* 

I.I2628 

3-038 

1.18046 

4.409 

1.24976 

6.227 

1.33830 

8-653 

I-45I4I 

11.935 

80 

1.12863 

2.781 

I 

18427 

4.042 

I.2559I 

5.718 

1.34827 

7.966 

1.46790 

11.032 

81 

1.13078 

2.519 

1.18777 

3.666 

1.26158 

5-195 

1-35754 

7-254 

I-48345 

10.081 

82 

I.I3273 

2.251 

I 

19094 

3-281 

1.26674 

4-656 

1.36607 

6.517 

1.49790 

9.087 

83 

I-I3447 

1.979 

I 

19377 

2.888 

1.27138 

4.104 

1.37378 

5-757 

1.51110 

8.052 

84 

I.I3598 

1.704 

1.19626 

2.488 

1-27545 

3-540 

1.38059 

4-975 

1.52289 

6-977 

85 

.13727 

1.425 

I 

19839 

2.083 

1.27895 

2.966 

1.38646 

4-175 

I-533I4 

5-868 

86 

.13832 

1.144 

I 

20013 

1.672 

I.28l85 

2.383 

1.39132 

3-360 

1.54170 

4-731 

87 

.13915 

0.860 

I 

20150 

1.257 

1.28413 

1.794 

1.39518 

2.532 

1.54848 

3-571 

88 

•13975 

0-574 

I 

.20247 

0.839 

I-28575 

1.199 

1-39793 

1.692 

1-55339 

2.391 

89 

.I40IO 

0.287 

I 

.20306 

0.420 

I.2867I 

0.600 

1-39959 

0.847 

I-55637 

1.198 

90      1.14022    o.ooo      1.20327    o.ooo      1.28700     o.ooo      1.40017     o.ooo      1.55740     o.ooo 


Note. 


Negative  quantities  are  in  heavy  type. 


-49' 


[27] 


tanh  6 


1ABLE    V. 
I.I 

^UK.IVJl,l^  1  UN  U 
1.2 

0 

—  r  /j.     i_or 
1.4 

JTINUEU 
1-5 

0 

o 

o 

o 

0 

o 

45 

0.90354     2I.I67 

0.87142     24.384 

0.83515  27.536 

0.79593  30.516 

0.75487     33.288 

46 

0.91359     21.524 

0.88183     24.856 

0.84561   28.118 

0.80600   31.225 

0.76427     34.116 

47 

0.92400     21.869 

0.89275     25.315 

0.85661    28.699 

0.81664   31.936 

0.77420     34-952 

48 

0.93482     22.2OI 

0.90425     25.764 

0.86815   29.279 

0.82786   32.648 

0.78467     35-792 

49 

0.94618     22.52O 

0.91625    26.2O2 

0.88038  29.852 

0.83971  33.360 

0-79573     36.638 

50 

0.95809     22.822 

0.92892     26.629 

0.89331   30.416 

0.85228   34.070 

0.80747     37-490 

0.97055     23.109 

0.94225     27.043 

0.90685  30.973 

0.86557  34-778 

0.81993     38.348 

52 

0.98345     23.379 

0.95617    27.442 

0.92123    31.521 

0.87964   35.485 

0.83320     39-213 

53 

0.99700     23.630 

0.97083     27.826 

0.93638  32.059 

0.89457  36.189 

0.84727     40.086 

54 

i  .01  1  r      23.861 

0.98625    28.193 

0.95238  32.586 

0.91043    36.890 

0.86220     40.965 

55 

1.0258        24.O7O 

1.0025       28.540 

0.96938   33.100 

0.92729  37.587 

0.87813     41.849 

56 

.O4I2         24.256 

1.0196       28.867 

0.98738    33-599 

0.94529  38.279 

0.89513     42.738 

57 

.0572         24.417 

1-0375       29.173 

1.0065      34-082 

0.96443  38.964 

0.91333     43.634 

58 

.0739         24.551 

1.0564       29.454 

1.0266      34.548 

0.98479  39.642 

0.93280    44-534 

59 

.0914         24.657 

1.0763       29.708 

1.0482      34-993 

1.  0066     40.311 

0-95367    45-441 

6o 

.1096         24.733 

1.0973       29.932 

1.0710      35.416 

1.0301      40.968 

0.97613    46.349 

61 

.1286         24.774 

I.II95       30.124 

1-0953      35-8x3 

1.0551      41.611 

1.0003      47-262 

62 

.1484        24.778 

1.1428       30.280 

I.I2I2        36.180 

1.0820     42.239 

1.0264      48.176 

63 

1.1689         24.745 

1.1674       30-397 

I.I488        36.515 

.1109     42.850 

1.0546      49.092 

64 

I.I9O4        24.672 

I.I934       30.473 

1.1784        36.814 

.1421     43.438 

1.0851      50.009 

65 

I.2I26         24.554 

1.  2208       30.502 

1.2099        37.072 

.1759    44.001 

1.1184      50.924 

66 

1.2357         24.388 

1.2496       30.480 

1.2436        37.285 

.2124    44.535 

1.1546      51-834 

67 

1.2596         24.173 

1.2799       30.403 

1.2798        37.446 

.2520    45.034 

1.1943      52.739 

68 

1.2845         23.904 

1.3119       30.263 

i-3l85      37-548 

.2951      45-493 

1-2377      53.636 

69 

I.3IOI         23.579 

1-3457      30-059 

1.3600     37-585 

•3420      45-904 

1.2857      54.520 

70 

1.3365         23.194 

1.3811      29.781 

1.4046      37-548 

•3933      46.260 

I-3385      55.388 

1.3637         22.744 

1.4183      29.425 

1-4525      37-430 

.4495      46.552 

1-3973      56.235 

72 

I.39I5         22.227 

1.4573      28.983 

1-5039      37-218 

.5113      46.768 

1.4628      57.056 

73 

1.4200        21.639 

1.4981      28.447 

1.5592      36.899 

1.5794     46.894 

1-5362      57-841 

74 

1.4489         20.978 

1.5407      27.810 

1.6186      36.461 

1.6546      46.916 

1.6190      58.581 

75 

1.4783         20.238 

1.5849      27.064 

1.6822      35.890 

1.7380     46.814 

1.7129      59.266 

76 

1.5077         19.419 

1.6307      26.202 

1-7505      35.170 

1.8308      46.567 

1.8202      59.880 

77 

1.5373         I8.5I8 

1.6778      25.214 

1.8235      34-280 

1.9341      46.145 

1.9438      60.403 

78 

1.5665         17.532 

1.7261      24.094 

1.9012      33.202 

2.0498     45.514 

2.0875      60.814 

79 

1.5955         16.462 

1.7749      22.835 

1.9835      31.912 

2.1792      44-633 

2.2562      61.076 

80 

1.6235         15.308 

1.8239      21.429 

2.0702      30.389 

2.3240      43-454 

2.4563      61.144 

81 

1.6505         14.069 

1.8726      19.874 

2.1604      28.609 

2.4862      41-915 

2.6973      60.956 

82 

1.6759         12.748 

1.9200      18.168 

2.2531      26.553 

2.6669      39.944 

2.9917      60.422 

83 

1.6996         11.350 

1.9654      16.310 

2.3466      24.200 

2.8668      37-454 

3-3577      59-412 

84 

I.72I2            9.879 

2.0078      14.306 

2.4388      21.540 

3.0848      34-354 

3.8207      57.724 

85 

1.7402            8.343 

2.0463      12.164 

2.5267      18.568 

3.3164      30.541 

4.4198      55-045 

86 

1.7562            6.748 

2.0796        9.898 

2.6065      I5-294 

3.5521      25.931 

5.2010     50.866 

87 

1.7691            5.105 

2.1068        7.525 

2.6744      11.747 

3.7765      20.477 

6.2275      44-356 

88 

1.7785            3.425 

2.1269        5-067 

2.7266        7.970 

3.9665      14.214 

7.5060     34-274 

89 

1.7843            I.7I9 

2-1393        2.548 

2.7596        4.029 

4.0952        7.293 

8.7880      19.305 

90 

1.7862        o.ooo 

2.1434        o.ooo 

2.7709        o.ooo 

4.1413        o.ooo 

9.4007        o.ooo 

Note. 

Negative  quantities  are  in  heavy  type. 

tanh  (1.3  /45°) 

Example. 

i-3  /45° 

0.83515  \27°.536  = 

-  0.83515  \i7°.32' 

.10". 

[28] 


TABLE  V. 

CORRECTING 

^.™^   tanhfl 

INTJED 

1-AC        ^   -  r£y.  CONT 

o 

45 
46 
47 
48 

49 

1.6 

o 

0.71331  35-799 
0.72175  36.733 
0.73069  37.682 
0.74013  38.642 
0.75013  39-6i5 

o 

0.67224  38.016 

0.67959  39-048 

0.68741  40.098 
0.69565  41.166 

0.70435  42.251 

1.8 

0 

0.63250  39-937 
0.63883  41.050 
0.64544  42.186 
0.65239  43-341 
0.65978  44.520 

1.9 

o 

0.59484  41.562 

0.60005  42.740 
0.60553  43.942 
0.61126  45.168 
0.61726  46.422 

2.0 
o 
0-55955  42-906 
0.56375  44-133 
0.56815  45-387 
0.57275  46.667 
0.57750  47-977 

50 

52 
53 
54 

0.76069  40.600 
0.77194  41.600 
0.78381  42.616 
0.79650  43.646 
0.81000  44.691 

0.71359  43-355 
0.72329  44.480 

o.73365  45-627 
0.74465  46.797 
0.75629  47-992 

0.66756  45.725 
0.67572  46.955 
0.68433  48.213 
0.69350  49.501 
0.70317  50.820 

0.62358  47.705 

0.63026  49.019 
0.63721  50.366 
0.64458  51.747 
0.65232  53-165 

0.58250  49-319 
0.58770  50.694 
o-593I5  52-103 
0.59885  53-550 
0.60480  55-039 

55 
56 
57 
58 
59 

0.82438  45-754 
0.83969  46.834 
0.85613  47-931 
0.87369  49.047 
0.89256  50.183 

0.76865  49.212 
0.78188  50.461 
0.79600  51.739 
0.81106  53.047 
0.82724  54.389 

0.71344  52.174 
0.72433  53-564 
0-73594  54-992 
0.74828  56.460 
0.76144  57-974 

0.66047  54.621 
0.66911  56.121 
0.67821  57.666 
0.68784  59.259 
0.69805  60.905 

o.6noo  56.571 
0.61750  58.149 
0.62435  59-777 
0.63150  61.458 
0.63905  63-195 

60 
61 
62 

63 
64 

0.91288  51.339 
0-93475  52.516 
0.95834  53.714 
0.98394  54-935 
1.0117   56.181 

0.84459  55-768 
0.86324  57.181 
0.88335  58.635 
0.90506  60.131 
0.92853  61.674 

0-77550  59-535 
0.79056  61.145 
0.80672  62.810 
0.82406  64.533 
0.84272  66.318 

0.70895  62.606 
0.72047  64.366 
0.73274  66.191 
0.74579  68.085 
0.75974  70.052 

0.64695  64.995 
0.65530  66.860 
0.66405  68.793 
0.67330  70.802 
0.68310  72.891 

65 
66 
67 
68 
69 

1.0419   57-452 
.0749   58.748 
.mi   60.072 
.1508   61.425 
.1948   62.809 

0.95400  63.266 
0.98176  64.910 

I.OI2I    66.6l3 
1.0453    68.376 
I.o8l7    7O.2O9 

0.86283  68.172 
0.88461  70.098 
0.90817  72.104 
0.93372  74-I96 
0.96156  76.382 

0.77463  72.099 
0.79063  74.232 
0.80774  76.459 
0.82611  78.786 

0.84579  8I.22I 

0.69340  75.068 
0.70430  77-338 
0.71585  79-707 
0.72810  82.184 
0.74105  84.776 

70 

72 
73 
74 

.2433   64.224 
.2974   65.674 
.3580   67.160 
.4262   68.686 
•5033   70.256 

I.I2I9    72.II6 
1.1663    74.IO6 
I.2I57    76.187 
1.2708    78.371 
1.3325    80.670 

0.99189  78.671 
.0251   81.074 
.0614   83.602 
.1013   86.266 
.1452   89.084 

0.86705  83.773 
0.88989  86.454 
0.91453  89.273 
O.94III  92.244 

0.96974  95-379 

0-75475  87.490 
0.76925  90.337 
0.78465  93-326 
0.80095  96.466 
0.81815  99.769 

75 
76 

77 
78 
79 

.5913   71-873 
.6924   73-543 
.8098   75-274 
J-9473   77-078 
2.1104   78.966 

1.4020   83.IOI 
1.4807   85.685 
1.5704   88.447 
1.6731    91.420 
I.79I5    94-644 

.1936   92.076 
.2470   95.261 
1.3061   98.664 
1.3714  102.315 
1.4436  106.250 

1.0005   98.692 

1.0337  IO2.2O2 
1.0693  105.925 
I.I074  IO9.88I 
I.I478  114.088 

0.83625  103.245 
0.85525  106.905 
0.87515  110.760 
0.89585  114.820 
0.91720  119.093 

80 
81 
82 

83 
84 

2.3068   80.960 
2.5472   83.087 
2.8474   85.390 
3.2326   87.937 
3-7435   90.839 

1.9288   98.175 
2.0894  IO2.O83 
2.2775  106.466 
2.4992  111.446 
2.7604  II7.I99 

1.5231  110.510 
1.6104  115.140 
1.7056  120.191 
1.8078  125.722 
1.9155  131-793 

I.I906  II8.568 
1.2352   123.344 
I.28l3  128.428 
I-3279  I33.842 

1-3739  I39-590 

0.93905  123.590 
0.96110  128.319 
0.98305  133-281 
1.0045  138.474 
1.0249  143.894 

85 
86 

87 
88 
89 

4.4502   94.289 
5.4825   98.668 
7.1175  104.791 
9.9938  114.760 
15.528  135.011 

3.0648  123.945 
34II4  I3L956 
3.7856  I4L527 
4.1472  152.868 
4.4231   165.896 

2.0255  138.448 
2.1329  145-720 
2.2309  153.606 
2.3107  162.043 
2.3628  170.905 

1.4179  145-673 
1.4579  152-075 
1.4921  158.768 
1.5184  165.700 
1.5349  172.805 

1.0436  149-528 
1.0601  155-359 
1.0738  161.359 
1.0840  167.494 
1.0904  173-723 

90 

21.395  180.000 

4.5275   iSo.OOO 

2.3813  180.000 

1.5406  180.000 

1.0925  180.000 

Note. 
Example. 

Negative  quantiti 
tanh  (2.0  /8o°) 

es  are  in  heavy  type. 

0.93905  \i23°.590 

=  0.93905  \  1  23°.35' 

24  • 

2.0  /8o° 

[29] 

TABLE  V. 

2.1 

CORRECTING 

2.2 

FACTOI 

2-. 

>    tanh0     _/- 

CONTINUED 
I          2.5 

6 

j 

-  r/j. 

2.i 

0 

o 

o 

0 

o 

0 

45 

0.52690 

43.993 

0.49691 

44.845 

0.46952 

45.492 

0.44467 

45.961 

O.422I2 

46.283 

46 

0.53019 

45.254 

0.49941 

46.127 

0.47139 

46.783 

0.44592 

47.252 

0.42292 

47.564 

47 

0.53362 

46.542 

O.5O2OO 

47.436 

0.47322 

48.101 

0.44717 

48.567 

0.42368 

48.868 

48 

0.53714 

47.859 

0.50465 

48.774 

0.47509 

49-446 

0.44842 

49.908 

0.42440 

50.195 

49 

0.54081 

49.206 

0.50732 

50.142 

0.47696 

50.820 

0.44958 

51.277 

0.42504 

5L548 

5° 

0.54462 

50.586 

O.5IOO9 

51.542 

0.47883 

52.226 

0.45075 

52.676 

0.42560 

52.929 

51 

0.54852 

52.001 

0.5I29I 

52.978 

0.48070 

53-666 

0.45188 

54.107 

0.42612 

54-340 

52 

0.55257 

53.452 

0.51578 

54.452 

0.48256 

55-144 

0.45292 

55.572 

0.42652 

55.782 

53 

0.55681 

54-943 

0.51868 

55.965 

0.48443 

56-659 

0.45392 

57.073 

0.42684 

57.256 

54 

0.56114 

56.476 

0.52168 

57.520 

0.48630 

58.215 

0.45488 

58.614 

0.42704 

58.766 

55 

0.56562 

58.055 

0.52473 

59-120 

0.48813 

59.8i5 

0.45571 

60.196 

0.42712 

60.314 

56 

0.57029 

59.682 

0.52782 

60.769 

0.48991 

61.462 

0.45646 

61.822 

0.42708 

61.903 

57 

0.57514 

61.361 

0.53095 

62.469 

0.49170 

63.160 

0.45713 

63.496 

0.42688 

63-534 

58 

0.58014 

63.094 

0.53414 

64.226 

0-49339 

64.912 

0.45767 

65.220 

0.42648 

65.211 

59 

0.58533 

64.888 

0.53736 

66.042 

0.49504 

66.722 

0.45804 

66.998 

0.42588 

66.938 

60 

0.59071 

66.745 

0.54064 

67.921 

0.49661 

68.594 

0.45829 

68.835 

0.42512 

68.718 

61 

0.59629 

68.671 

0-54395 

69.869 

0.49813 

70.531 

0.45838 

70.735 

0.42412 

70.556 

62 

0.60210 

70.669 

0.54732 

71.889 

0-49957 

72.540 

0.45829 

72.701 

0.42288 

72.455 

63 

0.60814 

72.744 

0.55073 

73-988 

0.50087 

74.625 

0.45800 

74.740 

O.42I4O 

74.419 

64 

0.61443 

74.902 

0.55418 

76.171 

O.5O2O9 

76.791 

0.45750 

76.855 

0.41960 

76-455 

65 

0.62090 

77.151 

0.55764 

78.443 

0.50317 

79-044 

0.45675 

79.053 

0.41752 

78.567 

66 

0.62767 

79.496 

0.56109 

80.810 

0.50413 

81.390 

0-45579 

81.341 

O.4I5I2 

80.761 

67 

0.63471 

81.942 

0.56459 

83.280 

0.50491 

83.836 

0-45454 

83.723 

O.4I24O 

83.044 

68 

0.64205 

84.499 

0.56814 

85.858 

0.50557 

86.389 

0.45300 

86.208 

0.40928 

85.424 

69 

0.64962 

87.172 

0.57164 

88.552 

0.50604 

89.056 

0.45I2I 

88.802 

0.40580 

87.906 

70 

0.65752 

89.968 

0.57518 

91-370 

0.50630 

91.843 

0.44913 

91-514 

0.40196 

90.5OO 

0.66571 

92.898 

0.57873 

94-319 

0.50639 

94-759 

0.44671 

94-352 

0.39771 

93-215 

72 

0.67424 

95-968 

0.58223 

97.405 

0.50626 

97.811 

0.44396 

97-324 

0.39306 

96.061 

73 

0.68305 

99.185 

0.58568 

100.636 

0.50596 

101.006 

0.44092 

100.439 

0.38802 

99.048 

74 

0.69214 

102.560 

0.58914 

104.020 

0.50543 

104.352 

0-43754 

103.706 

0.38260 

IO2.I87 

75 

0.70152 

106.099 

0.59255 

107.564 

0.50470 

107.858 

0.43388 

107.134 

0.37681 

105.489 

76 

0.71114 

109.811 

0.59586 

111.274 

0.50378 

111.531 

0.42992 

110.730 

0.37068 

108.966 

77 

0.72096 

113.702 

0.59914 

115-156 

0.50265 

"5-374 

0.42575 

114-505 

0.36427 

112.633 

78 

0.73086 

117.778 

0.60232 

119.212 

0.50135 

II9-392 

0.42138 

118.467 

0.35762 

II6.5OI 

79 

0.74090 

122.043 

0.60536 

123.446 

0.49991 

123.590 

0.41683 

122.620 

0.35080 

120.582 

80 

0.75086 

126.501 

0.60832 

127.859 

0.49835 

127.968 

O.4I22I 

126.968 

0.34392 

124.887 

81 

0.76067 

131-151 

O.6lIO9 

132.449 

0.49674 

132.527 

0.40759 

131-517 

0.33706 

129.427 

82 

0.77019 

135-989 

0.61364 

137-212 

0.49509 

137-264 

0.40305 

136.267 

0.33033 

I34.209 

83 

0.77924 

141.011 

O.6l6oO 

142.140 

0-49343 

142.171 

0.39869 

141.212 

0.32391 

I39.233 

84 

0.78767 

146.210 

0.6l8l4 

147-223 

0.49183 

I47.237 

0.39463 

146.344 

0.31790 

144.499 

85 

0.79524 

I5L570 

O.62OOO 

152.449 

0.49039 

152.450 

0.39094 

151.650 

0.31248 

149.994 

86 

0.80176 

157.070 

0.62155 

157.800 

0.48913 

157-793 

0.38775 

157.113 

0.30778 

I55-703 

87 

0.80710 

162.690 

O.62282 

163.254 

0.48804 

163.245 

0.38516 

162.710 

0.30396 

161.598 

88 

0.81100 

168.406 

0.62374 

168.790 

0.48726 

168.781 

0.38325 

168.413 

0.30II3 

167.645 

89 

0.81338 

174.187 

O.62427 

174-381 

0.48678 

I74.376 

0.38207 

174.188 

0.29940 

173.796 

90 

0.81424 

180.000 

0.62445 

180.000 

0.48661 

180.000 

0.38167 

180.000 

0.29880 

iSo.OOO 

Note 

Negative  quantities  are  in  heavy  type. 

0 

1  -  Atllll^l  C  • 

2-3  /9Q° 

"•  \I50 

=  0.48661 

Zi_^ 

[30] 

1 

TABLE  V. 

CORRECTING 

FACTOR   tanh  6   -  I-   ^~ 

NUED 

JT/V^  J.  VJJS-.       - 

0 

r  /~y.   \^<JJN  11 

0 

45 
46 
47 
48 

49 

2.6 
o 

0.40173  46.480 
0.40215  47-744 
0.40250  49.030 
0.40281  50.337 
0.40304  51.667 

.  2.7 

o 

0.38326  46.576 
0.38341  47.821 
0.38348  49.083 
0.38344  50.365 
0-38333  51-666. 

2.8 

o 

0.36657  46.595 

0.36650  47.816 
0.36632  49-051 
0.36607  50.303 

0.36571  51-572 

2.9 

o 
0.35145   46.554 
0.35121   47-748 
0.35086   48.955 
0.35045   50-175 
0-34997   5I-409 

3-0 

0 

0.33770  46.468 

0-33733   47-637 
0.33690   48.815 
0.33637   50.002 
0-33573   51-198 

50 
5i 

52 

53 
54 

0.40319  53.022 
0.40319  54-403 
0.40308  55.812 
0.40288  57.250 
0.40254  58-717 

0.38311  52.988 
0.38278  54-332 
0.38233  55.700 
0.38174  57-094 
0.38096  58.513 

0.36525  52.857 

0.36464  54.160 

0.36389  55.483 
0.36300  56.826 
0.36196  58.190 

0.34928   52.656 
0.34848   53-9I6 

0-34759   55-I9I 
0.34652   56.481 
0.34528   57.788 

0-33497   52-405 
0.33410   53.622 

0-33307   54-849 
o-33  J91   56-086 
0-33058   57-334 

55 
56 

57 
58 
59 

0.40200  60.219 
0.40131  61.756 
0.40046  63.331 
0.39942  64.947 
0.39812  66.606 

0.38000  59-959 
0.37889  61.437 
0.37756  62.945 
0.37600  64.487 
0.37422  66.066 

0.36075  59-577 
0.35932  60.987 
0.35768  62.422 
o.3558o  63.884 
0.35367  65.374 

0.34386   59-"2 
0.34226   60.452 
0.34043   61.810 
0.33837   63.186 
0.33602   64.582 

0.32907   58.593 
0.32737   59-862 
0.32546   61.141 
0.32332   62.432 
0.32093   63.735 

6o 
61 
62 
63 
64 

0.39658  68.313 
0.39481  70.069 
0.39273  7I-879 
0.39035  73-747 
0.38765  75-677 

0.37215  67-684 
0.36980  69.343 
0.36716  71.048 
0.36416  72.801 
0.36081  74.607 

0.35126  66.895 
0.34856  68.449 
0.34554  70.036 
0.34217  71.660 
0-33843  73.325 

0-33345   65-999 
0.33056   67-439 
0.32734   68.904 
0.32378   70.392 
0.31985   71-907 

0.31827   65.049 
0.31532   66.375 
0.31206   67.713 
0.30847   69.063 
0.30451   70.426 

65 
66 
67 
68 

69 

0.38458  77-675 
0.38115  79-747 
0-37733  81.898 
0.37310  84.136 
0.36844  86.469 

0.35709  76.470 
0.35296  78.393 
0.34841  80.383 
0.34341  82.449 
0-33794  84.597 

0.33430  75-033 
0.32974  76.789 
0.32474  78.599 
0.31927  80.464 
0.31331  82.396 

0.31552   73.452 
0.31076   75-027 
0-30556   76-639 
0.29988   78.289 
0.29369   79.982 

0.30016   71.801 
0.29539   73.I92 
0.29018   74.598 
0.28450   76.021 
0.27832   77-463 

70 

7i 

72 

73 
74 

0.36333  88.902 
0-35777  9I-45I 
0.35176  94-122 
0.34529  96.929 
0-33837  99-884 

0.33198  86.833 
0.32553  89.168 
0-31857  91.613 
0.31109  94.184 
0.30311  96.893 

0.30683  84.399 
0.29982  86.482 
0.29226  88.655 
0.28415  90.931 
0.27548  93-325 

0.28697   81.722 
0.27970   83.518 
0.27186   85.378 
0.26343   87.310 
0.25439   89.328 

0.27161   78.926 
0.26434   80.414 
0.25649   81.932 
0.24803   83.485 
0.23894   85.081 

75 
76 
77 
78 
79 

0.33102  103.003 
0.32328  106.302 
0.31519  109.799 
0.30680  113.513 
0.29821  117.466 

0.29464  99-757 
0.28570  102.799 
0.27634  106.038 
0.26661  109.505 
0.25659  113.231 

0.26625  95.857 
0.25648  98.547 
0.24621  101.421 
0.23546  104.512 
0.22431  107.861 

0.24475   91.448 
0.23450   93.690 
0.22366   96.078 
0.21224   98.647 
0.20030  101.437 

0.22922   86.729 
0.21883   88.443 
0.20779   90.241 
0.19610   92.146 
0.18376   94.191 

80 
81 
82 

83 
84 

0.28950  121.678 
0.28081  126.171 
0.27227  130.966 
0.26405  136.078 
0.25633  141-517 

0.24637  117.249 
0.23609  121.599 
0.22589  126.322 
0.21597  131.464 
0.20654  137.060 

0.21284  111.519 
0.20116  115.543 
0.18943  120.004 
0.17783  124.989 
0.16660  130.591 

0.18788  104.504 
0.17507  107.921 
0.16200  111.784 
0.14880  116.221 
0-13573  121.403 

0.17082   96.419 
0.15730   98.893 
0.14329  101.700 
0.12887  104.976 
0.11419  108.916 

85 
86 

87 
88 
89 

0.24931  147.286 
0.24319  153-370 
0.23818  159.740 
0.23446  166.349 
0.23216  173.129 

0.19786  143-143 
0.19020  149.722 
0.18384  156.778 
0.17906  164.253 
0.17609  172.042 

0.15604  136.907 
0.14651  144.025 
0.13844  151.988 
0.13224  160.762 
0.12832  170.195 

0.12307  127.549 

O.III22   134.931 
0.10074   I43-833 
0.092383  I54-456 
0.086890  166.709 

0.099443  113-839 
0.084970  120.254 
0.071310  128.990 
0-059383  141.294 
0.050750  158.480 

90 

0.23138  180.000 

0.17509  180.000 

0.12698  180.000 

0.084969  I8O.OOO 

0.047517  180.000 

Note. 
Example. 

Negative  quantities  are  in  heavy  type, 
tanh  (2.9  /85°)         ,  „  t   a  ., 

2-9/85! 

=  0.12307  \i27  .549 

-  0.12307  \i27--32 

.56'. 

[31] 


TABLE  VI 


FUNCTIONS  OF  SEMI-IMAGINARIES.    /(p/4£)  =  r  fa 
p                       Sinh                                  Cosh                               Tanh 

0      / 

0      / 

o     / 

o 

o. 

45-00 

I. 

0. 

o. 

45-00 

O.I 

O.IOOOO 

45.06 

I.OOOOI 

0.17 

0.09999 

44.49 

O.2 

O.2OOOO 

45-23 

1.00013 

1.09 

0.19997 

44.14 

o-3 

0.30001 

45-52 

1.0007 

2-35 

0.29981 

43-17 

0.4 

0.40005 

46.32 

1.  002  1 

4-35 

0.39921 

41-57 

o-S 

0.50016 

47-23 

1.0052 

7.08 

0-49757 

40.15 

0.6 

0.60042 

48.27 

I.OIO7 

10.15 

0.59406 

38.11 

0.7 

0.70094 

49.40 

I.OigS 

13-53 

0.68732 

35-47 

0.8 

0.80184 

51.06 

1.0336 

18.00 

0-77577 

33-o6 

0.9 

0.90327 

52.44 

1-0533 

22.34 

0.85756 

30.10 

I.O 

1-0055 

54-32 

1.0803 

27.29 

0.93077 

27.03 

i.i 

1.1089 

56.31 

i."57 

32-41 

0.99389 

23-50 

1.2 

1.2138 

58.41 

1.  1608 

38-05 

1-0457 

20.36 

i-3 

1.3205 

61.02 

1.2163 

43-35 

1.0857 

17.27 

1.4 

1.4297 

63-34 

1.2830 

49-05 

1.  1143 

14.29 

i-5 

1.5418 

66.15 

1.3616 

54-33 

1-1323 

11.42 

1.6 

1-6575 

69.07 

1-4524 

59-55 

1.1413 

9.12 

i-7 

1.7776 

72.08 

I.5556 

65.09 

1.1428 

6-59 

1.8 

1.9029 

75.18 

1.6714 

70.14 

1-1385 

5-04 

1.9 

2.0343 

78.36 

1.7999 

75-io 

1.1302 

3-26 

2.O 

2.1726 

82.01 

1-9413 

79.56 

1.1191 

2.06 

2.1 

2.3190 

85-34 

2.0958 

84-33 

1.1065 

I.OI 

2.2 

2-4745 

89.12 

2.2636 

89.03 

1.0932 

0.09 

2-3 

2.6404 

92.57 

2.4449 

93.26 

1.0799 

0.29 

2.4 

2.8177 

96.46 

2.6403 

97-44 

1.0672 

0.58 

2-5 

3.0079 

100.39 

2.8502 

101.56 

1-0553 

1.17 

2.6 

3.2121 

104.36 

3-0753 

106.05 

1.0445 

1.29 

2-7 

3.43i8 

108.36 

3-3!63 

IIO.IO 

1.0348 

1-34 

2.8 

3-6685 

112.39 

3-5741 

114.15 

1.0264 

1.36 

2.9 

3.9236 

116.43 

3-8497 

118.16 

1.0192 

1.33 

3-o 

4.1986 

1  20.48 

4-1443 

122.16 

1.0131 

1.28 

3-i 

4.4948 

124.56 

4.4589 

126.15 

1.0080 

1.19 

3-2 

4-8154 

129.02 

4-7955 

!30.i5 

1.0041 

1.13 

3-3 

5.1586 

133.09 

5.I54I 

I34-I3 

1.0008 

1.04 

3-4 

5-5306 

137.17 

5-5393 

138-13 

0.9984 

0.56 

3-5 

5-9305 

141.24 

5.9356 

142.12 

0.9967 

0.48 

3-6 

6.3603 

145-31 

6.3900 

146.11 

0.9954 

0.40 

3-7 

6.8244 

149.38 

6.8606 

150.10 

0.9947 

0.32 

3-8 

7-3228 

153.44 

7.3646 

154.09 

0-9943 

0.25 

3-9 

7.8590 

157.50 

7.9047 

158.10 

0.9942 

O.2O 

4.0 

8.4351 

161.57 

8.4831 

162.11 

0.9943 

O.I4 

4-i 

9-0535 

166.02 

9.1024 

166.12 

0.9946 

O.IO 

4.2 

9.7198 

170.07 

9.7704 

170.13 

0.9948 

O.O6 

4-3 

10.434 

174.11 

10.481 

174-15 

0-9955 

0.04 

4.4 

II.  201 

178.16 

11.246 

178.16 

0.9960 

O.OO 

Note.  Negative  quantities  are  in  heavy  type. 

Examples,     sinh  (1.7  /45°)  =  1.7776  /72°.o8'. 
tanh  (2.4  /45°)  =  1.0672  \o°.58'. 

[32] 


TABLE  VI 
FUNCTIONS  OF  SEMI-IMAGINARIES.   /(p/4^)  =  r /y.    CONTINUED 


Cosech 


Sech 


Coth 


0. 

00 

45-00 

i. 

o. 

oo 

45-oo 

O.I 

IO.OOOO 

45-06 

0.99999 

0.17 

IO.OOOO 

44-49 

0.2 

5.0000 

45-23 

0.99987 

1.09 

5.0008 

44.14 

o-3 

3-3333 

45-52 

0-9993 

2-35 

3-3355 

43-17 

0.4 

2-4997 

46.32 

0.9979 

4-35 

2.5050 

41.57 

0-5 

1.9984 

47-23 

0.9948 

7.08 

2.0096 

40.15 

0.6 

1.6654 

48.27 

0.9894 

10.15 

1.6830 

38.11 

0.7 

1.4268 

49.40 

0.9806 

13-53 

1-4549 

35-47 

0.8 

1.2471 

51.06 

0.9675 

18.00 

1.2890 

33-o6 

0.9. 

1.1070 

52.44 

0.9494 

22.34 

1.  1660 

30.10 

I.O 

0.9945 

54-32 

0.9256 

27.29 

1.0746 

27.03 

i.i 

0.9018 

56.31 

0.8963 

32.41 

i.  006  1 

23-50 

1.2 

0.8238 

58.41 

0.8614 

38.05 

0.9564 

20.36 

i-3 

0-7573 

61.02 

0.8222 

43-35 

0.9211 

17.27 

1.4 

0.6995 

63-34 

0.7793 

49-05 

0.8996 

14.29 

•5 

0.6486 

66.15 

0-7344 

54-33 

0.8831 

11.42 

.6 

0.6033 

69.07 

0.6885 

59-55 

0.8763 

9.12 

•7 

0.5625 

72.08 

0.6429 

65.09 

0.8751 

6-59 

.8 

0.5256 

75-18 

0.5981 

70.14 

0.8788 

5-04 

•9 

0.4916 

78.36 

0.55,56 

75-10 

0.8848 

3-26 

2.0 

0.4603 

82.01 

0-5151 

79.56 

0.8936 

2.06 

2.1 

0.4312 

85-34 

0.4772 

84.33 

0.9038 

I.OI 

2.2 

0.4041 

89.12 

0.4418 

89.03 

0.9147 

0.09 

2-3 

0.3788 

92-57 

0.4090 

93-26 

0.9260 

0.29 

2.4 

0-3549 

96.46 

0.3788 

97-44 

0.9370 

0.58 

2-5 

0-3325 

100.39 

0-3509 

101.56 

0.9476 

1.17 

2.6 

0.3114 

104.36 

0.3252 

106.05 

0-9574 

1.29 

2.7 

0.2914 

108.36 

0.3016 

IIO.  10 

0.9663 

1-34 

2.8 

0.2726 

112.39 

0.2798 

114.15 

0-9743 

.36 

2-9 

0.2549 

116.43 

0.2598 

118.16 

0.9812 

•33 

3-o 

0.2382 

120.48 

0.2413 

122.16 

0.9871 

.28 

3-i 

0.2225 

124.56 

0.2243 

126.15 

0.9920 

.19 

3-2 

0.2077 

129.02 

0.2085 

130.15 

0.9959 

•13 

3-3 

0.1939 

133-09 

0.1940 

134-13 

0.9992 

.04 

3-4 

o.i  808 

137.17 

0.1805 

138.13 

1.0016 

0.56 

3-5 

o.i  686 

141.24 

0.1681 

142.12 

•0033 

0.48 

3-6 

0.1572 

I45-3I 

0.1565 

146.11 

.0047 

0.40 

3-7 

0.1465 

149.38 

0.1458 

150.10 

•0053 

0.32 

3-8 

0.1366 

153-44 

0.1358 

154.09 

.0057 

0.25 

3-9 

0.1272 

157-50 

0.1265 

158.10 

.0058 

O.2O 

4.0 

0.1186 

161.57 

01179 

162.11 

.0057 

O.I4 

4.1 

0.1105 

166.02 

0.1099 

166.12 

.0054 

O.IO 

4-2 

0.1029 

170.07 

0.1024 

170.13 

1.0052 

O.O6 

4-3 

0.09584 

174.11 

0.09541 

174-15 

1.0045 

O.O4 

4-4 

0.08927 

178.16 

0.08892 

178.16 

1.0040 

o.oo 

Note.  Negative  quantities  are  in  heavy  type. 

Examples,     cosech  (2.0  745°)  =  0.4603  \82°.oi'. 
coth     (2.5  /45JO  =  0.9476/1°. 1 7'- 

[33] 


TABLE  VI 
FUNCTIONS  OF  SEMI-IMAGINARIES.   /  (p  /45°)  =  r  /y.    CONTINUED 


Sinh 


Cosh 


4-5 

12.026 

182.19 

4.6 

12.909 

186.23 

4-7 

13-858 

190.27 

4.8 

14.876 

194.30 

4-9 

15-968 

198-33 

S-o 

17.140 

202.36 

5-i 

18.397 

206.39 

5-2 

19-747 

210.42 

5-3 

21.195 

214-45 

5-4 

22.750 

218.48 

5-5 

24.418 

222.50 

5-6 

26.219 

226.53 

5-7 

28.141 

230.56 

5-8 

30.192 

234-59 

5-9 

32-405 

239.02 

6.0 

34-784 

243-05 

P 

Cosech 

0    / 

4-5 

0.08316 

182.19 

4-6 

0.07746 

186.23 

4-7 

0.07216 

190.27 

4.8 

0.06722 

194.30 

4-9 

0.06263 

198.33 

S-o 

0.05834 

202.36 

5-i 

0.05436 

206.39 

5-2 

0.05064 

210.42 

5-3 

0.04718 

214.45 

5-4 

0.04396 

218.48 

5-5 

0.04095 

222.50 

5-6 

0.03814 

226.53 

5-7 

0.03554 

230.56 

5-8 

0.03312 

234.59 

5-9 

0.03086 

239.02 

6.0 

0.02875 

243.05 

12.067 
12.948 

13.894 
14.909 
15-999 

17.169 

18.425 
19.772 

21.219 

22.772 

24.439 
26.238 
28.159 
30.209 
32.421 


182.19 
186.21 
190.23 
194.26 
198.29 

202.32 
206.35 
210.38 
214.41 

218.44 

222.47 
226.51 
230.54 
234.57 
239.00 


34.798  243.04 


Sech 


0.08288 
0.07723 
0.07197 
0.06707 
0.06250 

0.05824 
0.05428 
0.05058 
0.04713 
0.04391 

0.04092 
0.03811 
0-03551 
0.03310 
0.03085 


182.19 
186.21 
190.23 
194.26 
198.29 

202.32 
206.35 
210.38 
214.41 
218.44 

222.47 
226.51 
230.54 
234-57 
239.00 


Tanh 

O       I 

0.9966      o.oo 

0.9970  O.02 

0.9974  O.O4 

0.9978  O.O4 

0.9980  O.04 


0.9983 
0.9985 


O.O4 
O.04 


0.9087  O.O4 
0.9989  O.O4 
0.9990  O.O4 


0.9992 

0-9993 
0-9994 
0.9995 


0.03 

O.O2 
O.O2 
O.02 


0.9996          O.O2 

0.9996      o.oi 
Coth 


1.0034 

o.oo 

1.0030 
1.0026 

O.O2 
O.O4 

I.OO22 

O.O4 

I.OO2O 

O.O4 

I.OOI7 

O.O4 

I.OOI5 

O.O4 

I.OOI3 

O.O4 

I.OOII 

O.O4 

I.OOIO 

O.04 

1.0008 

0.03 

1.0007 
1.  0006 

O.O2 
O.O2 

1.0005 

O.02 

1.0004 

O.O2 

0.02874       243.04 


1.0004      o.oi 


Note.  Negative  quantities  are  in  heavy  type. 

Examples,     tanh  (6.0  745°)  =  0.9996  /o°.oi'. 

seen  (5.0  /45°)  =  0.05824  \2O2°.32'. 


[34] 


TABLE  VI 
FUNCTIONS  OF  SEMI-IMAGINARIES.   /(p/4s°)  =  r /y.    CONTINUED 


Sinh  and  cosh 


Tanh  and  coth 


Sech  and  cosech 


6.05 

36.047 

245-06 

I.OOO 

o.oo 

2.774) 

<IO~2 

245.06 

6.io 

37-349 

247.08 

I.OOO 

o.oo 

2.678 

u 

247.08 

6.iS 

38-693 

249.09 

I.OOO 

o.oo 

2.583 

• 

249.09 

6.20 

40.084 

251.11 

I.OOO 

o.oo 

2-495 

" 

251.11 

6.25 

41-524 

253-12 

I.OOO 

o.oo 

2.408 

u 

253.12 

6.30 

43-020 

255-I4 

I.OOO 

o.oo 

2.325 

a 

255.14 

6-35 

44-563 

257.15 

I.OOO 

o.oo 

2.244 

u 

257.15 

6.40 

46.171 

259-I7 

I.OOO 

o.oo 

2.166 

u 

259.17 

6-45 

47-832 

261.18 

I.OOO 

o.oo 

2.091 

u 

261.18 

6.50 

49-553 

263.20 

I.OOO 

o.oo 

2.018 

a 

263.20 

6.55 

5I-336 

265.22 

I.OOO 

o.oo 

1.948 

u 

265.22 

6.6o 

53-i83 

267.24 

I.OOO 

o.oo 

1.  880 

a 

267.24 

6.65 

55-no 

269.25 

I.OOO 

o.oo 

1.815 

u 

269.25 

6.70 

57-058 

271.27 

I.OOO 

o.oo 

1.752 

u 

271.27 

6-75 

59-I36 

273.28 

I.OOO 

o.oo 

1.691 

u 

273.28 

6.80 

61.259 

275-30 

I.OOO 

o.oo 

1.632 

u 

275.30 

6.85 

63-463 

277-31 

I.OOO 

o.oo 

1.576 

u 

277.31 

6.90 

65-746 

279-33 

I.OOO 

o.oo 

1.521 

u 

279.33 

6-95 

68.119 

281.34 

I.OOO 

o.oo 

1.468 

u 

281.34 

7.00 

70.570 

283-36 

I.OOO 

o.oo 

1.4*7 

a 

283.36 

7-05 

73.109 

285-37 

I.OOO 

o.oo 

1.368 

u 

285.37 

7.10 

75-739 

287.39 

I.OOO 

o.oo 

1.312 

u 

287.39 

7-iS 

78.473 

289.40 

I.OOO 

o.oo 

1.274 

a 

289.40 

7.20 

81.296 

291.42 

I.OOO 

o.oo 

1.230 

a 

291.42 

7-25 

84.215 

293-43 

I.OOO 

o.oo 

1.187 

u 

293.43 

7-30 

87.250 

295-45 

I.OOO 

o.oo 

1.146 

a 

295.45 

7-35 

90.386 

297.46 

I.OOO 

o.oo 

1.016 

u 

297.46 

7.40 

93-083 

299.48 

I.OOO 

o.oo 

1.074 

u 

299.48 

745 

97.009 

301.49 

I.OOO 

o.oo 

1.031 

a 

301.49 

7-5° 

100.50 

303-51 

I.OOO 

o.oo 

<IO~* 

303.51 

7-55 

104.12 

305-52 

I.OOO 

o.oo 

9.605 

u 

305-52 

7.60 

107.86 

307-54 

I.OOO 

o.oo 

9.271 

u 

307.54 

7-65 

111.74 

309-56 

I.OOO 

o.oo 

8.949 

u 

309.56 

7.70 

115.67 

3H-57 

I.OOO 

o.oo 

8.638 

u 

311.57 

7-75 

119.94 

3*3.59 

I.OOO 

o.oo 

8-337 

u 

313.59 

7.80 

124.26 

316.00 

I.OOO 

o.oo 

8.048 

u 

316.00 

7.85 

128.71 

318.02 

I.OOO 

o.oo 

7.769 

a 

318.02 

7.90 

133-35 

320.03 

I.OOO 

o.oo 

7-499 

a 

320.03 

7-95 

138.16 

322.05 

I.OOO 

o.oo 

7.238 

a 

322.05 

8.00 

143.12 

324-06 

I.OOO 

o.oo 

6.987 

u 

324.06 

8.05 

148.28 

326.07 

I.OOO 

o.oo 

6.744 

1C 

326.07 

8.10 

153.61 

328.09 

I.OOO 

o.oo 

6.510 

u 

328.09 

8.15 

I59-I4 

330.11 

I.OOO 

o.oo 

6.284 

a 

330.11 

8.20 

164.87 

332-12 

I.OOO 

o.oo 

6.066 

u 

332.12 

8.25 

170.80 

334-14 

I.OOO 

o.oo 

5-855 

a 

334.14 

Note.  Negative  quantities  are  in  heavy  type. 

Examples,     sinh  (7.55  /45°)  =  cosh  (7.55  /45°)  =  104.12 


sech  (7-50/45!)  =  cosech  (7.50/45!)  =  9-95Q  X  io-»\303°.si' 

[35] 


TABLE  VI 
FUNCTIONS  OF  SEMI-IMAGINARIES.    f  (p  [*£)  =  r 


CONTINUED 


Sinh  and  cosh 


Tanh  and  coth 


Sech  and  cosech 


8.30 

176.95 

336.15 

I.OOO 

o.oo 

5.651X10-3 

336.15 

8-35 

183.31 

338.17 

I.OOO 

o.oo 

5-455 

• 

338.17 

8.40 

189.91 

340.18 

I.OOO 

o.oo 

5.266 

« 

340.18 

8.45 

196.75 

342.20 

I.OOO 

o.oo 

5.083 

a 

342.20 

8.50 

203.83 

344-22 

I.OOO 

o.oo 

4.906 

(( 

344-22 

8-55 

211.  l6 

346.24 

I.OOO 

o.oo 

4.736 

« 

346.24 

8.60 

218.76 

348.25 

I.OOO 

o.oo 

4.571 

« 

348.25 

8.65 

226.63 

350.27 

I.OOO 

o.oo 

4.413 

« 

350.27 

8.70 

234-79 

352.28 

I.OOO 

o.oo 

4.259 

H 

352.28 

8-75 

243.23 

354-30 

I.OOO 

o.oo 

4.111 

a 

354.30 

8.80 

251-99 

356.31 

I.OOO 

o.oo 

3.968 

« 

356.31 

8.85 

261.06 

358.33 

I.OOO 

0.00 

3-830 

It 

358.33 

8.90 

270.46 

360.34 

I.OOO 

o.oo 

3.698 

u 

360.34 

8-95 

280.19 

362.36 

I.OOO 

o.oo 

3.569 

u 

362.36 

9.00 

290.28 

364-38 

I.OOO 

o.oo 

3-445 

" 

364.38 

9-05 

300.73 

366.39 

I.OOO 

o.oo 

3.3253 

• 

366.39 

9.10 

3H.54 

368.41 

I.OOO 

o.oo 

3.2099 

u 

368.41 

9-i5 

322.75 

370.42 

I.OOO 

o.oo 

3.0983 

u 

370.42 

9.20 

334-37 

372-44 

I.OOO 

o.oo 

2.9908 

u 

372.44 

9-25 

346.39 

37446 

I.OOO 

o.oo 

2.8869 

u 

374.46 

9-3° 

358.85 

376.47 

I.OOO 

o.oo 

2.7867 

u 

376.47 

9-35 

371.81 

378.48 

I.OOO 

o.oo 

2.6895 

a 

378.48 

9.40 

385-15 

380.50 

I.OOO 

o.oo 

2.5964 

u 

380.50 

9-45 

399.04 

382.51 

I.OOO 

o.oo 

2.5060 

u 

382.51 

9-5° 

413-38 

384.53 

I.OOO 

o.oo 

2.4191 

" 

384.53 

9-55 

428.26 

386.55 

I.OOO 

o.oo 

2-3350 

u 

386.55 

9.60 

443.67 

388.56 

I.OOO 

o.oo 

2.2540 

u 

388.56 

9-65 

446.93 

390-57 

I.OOO 

o.oo 

2.2263 

u 

390.57 

9.70 

476.18 

392.59 

I.OOO 

o.oo 

2.IOOI 

u 

392.59 

9-75 

493-31 

395-01 

I.OOO 

o.oo 

2.0271 

u 

395-01 

9.80 

511.07 

397.02 

I.OOO 

o.oo 

1.9567 

u 

397.02 

9-85 

529.46 

399-03 

I.OOO 

o.oo 

1.8887 

u 

399.03 

9.90 

548.52 

401.05 

I.OOO 

o.oo 

1.8231 

u 

401.05 

9-95 

568.25 

403.07 

I.OOO 

o.oo 

1.7598 

u 

403.07 

10.00 

588.69 

405.08 

I.OOO 

0.00 

r.6987 

u 

405.08 

10.05 

609.89 

407.09 

I.OOO 

o.oo 

1.6397 

u 

407.09 

IO.IO 

631.84 

409.11 

I.OOO 

0.00 

1.5827 

u 

409.11 

10.15 

654-58 

411.13 

I.OOO 

o.oo 

•5277 

u 

411.13 

IO.2O 

678.14 

4I3-H 

I.OOO 

o.oo 

•4746 

u 

413-14 

10.25 

702.53 

415-15 

I.OOO 

o.oo 

•4234 

a 

415.15 

10.30 

727.81 

417.17 

I.OOO 

o.oo 

•3740 

u 

417.17 

10-35 

754-01 

419.19 

I.OOO 

o.oo 

.3262 

u 

419.19 

10.40 

781.14 

421.21 

I.OOO 

o.oo 

.2802 

u 

421.21 

10.45 

809.26 

423-23 

I.OOO 

o.oo 

•2357 

a 

423-23 

10.50 

838.38 

425-24 

I.OOO 

o.oo 

1.1928 

u 

425.24 

Note.     Negative  quantities  are  in  heavy  type. 
Examples. 

sinh  (10.0/45^)  =  cosh  (10.0/45^)  =  588.69  /405°.o8'. 

sech  (IO.Q  /45°)  =  cosech  (10.0/45^)  =  1.6987  X  io~3  \4os°.o8'  =  1.6987  X  io~3  \45°.o8' 

[36] 


TABLE  VI 
FUNCTIONS  OF  SEMI-IMAGINARIES.  f(pl*£)  =  r  fa.    CONTINUED  ; 


Sinh  and  cosh 


Tanh  and  coth 


Sech  and  cosech 


10.55 

868.56 

427.26 

I.OOO 

o.oo 

1.  1513X10-' 

427.26 

io.6o 

899.81 

429.27 

I.OOO 

o.oo 

1.1113 

• 

429.27 

10.65 

932.18 

43L29 

I.OOO 

o.oo 

1.0728 

• 

431.29 

10.70 

965.74 

433-30 

I.OOO 

o.oo 

1.0555 

* 

433.30 

10.75 

1,000.5 

435-32 

I.OOO 

o.oo 

do"4 

435.32 

10.80 

1,036.5 

437-33 

I.OOO 

o.oo 

9.6478 

u 

437-33 

10.85 

1,073.8 

439-35 

I.OOO 

o.oo 

9.3128 

u 

439-35 

10.90 

1,112.4 

441-36 

I.OOO 

o.oo 

8.9892 

u 

441.36 

10.95 

1,152.5 

443-38 

I.OOO 

o.oo 

8.6770 

" 

443.38 

II.OO 

1,194.0 

445-39 

I.OOO 

o.oo 

8.3750 

" 

445.39 

11.05 

1,237.0 

447.41 

I.OOO 

o.oo 

8.0845 

u 

447.41 

II.  IO 

1,281.5 

449.42 

I.OOO 

o.oo 

7.8037 

u 

449.42 

11.15 

1,327.5 

451-44 

I.OOO 

o.oo 

7-5327 

a 

451.44 

II.  20 

i,375-3 

45346 

I.OOO 

o.oo 

7.2711 

u 

453.46 

11.25 

1,424.8 

455-47 

I.OOO 

o.oo 

7.0184 

u 

455-47 

11.30 

1,476.1 

457-48 

I.OOO 

o.oo 

6.7747 

u 

457.48 

"•35 

1,529-2 

459-50 

I.OOO 

o.oo 

6.5393 

a 

459.50 

11.40 

1,584-3 

561-52 

.000 

o.oo 

6.3120 

u 

461.52 

"•45 

1,641.4 

463-53 

.000 

o.oo 

6.0929 

a 

463.53 

11.50 

1,700.3 

465-54 

.000 

o.oo 

5.8811 

a 

465.54 

"•55 

1,761.5 

467.56 

.000 

o.oo 

5.6769 

u 

467.56 

1  1.  60 

1,824.9 

469-57 

.000 

o.oo 

5-4797 

a 

469.57 

11.65 

1,890.6 

471-59 

I.OOO 

o.oo 

5-2893 

u 

471.59 

11.70 

1,958.6 

474.01 

I.OOO 

o.oo 

5-1056 

u 

474.01 

"•75 

2,029.1 

476.03 

I.OOO 

o.oo 

4.9282 

u 

476.03 

n.8o 

2,102.1 

478.04 

I.OOO 

o.oo 

4.7571 

a 

478.04 

11.85 

2,177.8 

480.05 

I.OOO 

o.oo 

4.5910 

u 

480.05 

11.90 

2,256.1 

482.07 

I.OOO 

o.oo 

4.4323 

u 

482.07 

"•95 

2,337-3 

484.09 

I.OOO 

o.oo 

4.2784 

u 

484.09 

I2.OO 

2,421.5 

486.10 

I.OOO 

o.oo 

4.1297 

u 

486.10 

12.05 

2,508.6 

488.12 

I.OOO 

o.oo 

3.9862 

u 

488.12 

12.10 

2,598.9 

490.14 

I.OOO 

o.oo 

3-8478 

a 

490.14 

12.15 

2,692.6 

492.15 

I.OOO 

o.oo 

3-7141 

u 

492.15 

12.  2O 

2,789.0 

494.17 

I.OOO 

o.oo 

.3-5856 

u 

494.17 

12.25 

2,889.7 

496.18 

I.OOO 

o.oo 

3-4605 

a 

496.18 

12.30 

2,993-7 

498.20 

I.OOO 

o.oo 

3-3403 

u 

498.20 

12-35 

3,101.4 

500.21 

I.OOO 

o.oo 

3-2243 

u 

500.21 

12.40 

3,213-1 

502-23 

I.OOO 

o.oo 

3-0143 

u 

502.23 

12-45 

3,328.3 

504.24 

I.OOO 

o.oo 

3.0042 

u 

504.24 

12.50 

3,448.5 

506.26 

I.OOO 

o.oo 

2.8998 

a 

506.26 

12.55 

3,572.6 

508.27 

I.OOO 

o.oo 

2.7991 

a 

508.27 

12.60 

3,7oi.i 

510.29 

I.OOO 

o.oo 

2.7019 

u 

510.29 

12.65 

3,834-3 

5I2-3i 

I.OOO 

o.oo 

2.6080 

a 

512.31 

12.70 

3,972.6 

514-32 

I.OOO 

o.oo 

2.5172 

u 

314.32 

12.75 

4,"5-3 

516.33 

I.OOO 

o.oo 

2.4300 

a 

516.33 

Note.  Negative  quantities  are  in  heavy  type. 

Examples,     sinh  (12.0  /45°)  =  cosh  (12.0/45°)  =  2421.5 


86°.  10'  =  2421.5 
sech  (12.75  /45°)  =  cosech  (12.75  /45°)  =  2.43  X  io-3\si6°.33'. 

[37] 


442282 


TABLE  VI 
FUNCTIONS  OF  SEMI-IMAGINARIES.   /  (p  /45!)  =  r  /y.    CONTINUED 


Sinh  and  cosh 


Tanh  and  coth 


Sech  and  cosech 


12.80 

4,263.4 

518.35 

I.OOO 

o.oo 

2-3455  X 

I0~4 

518.35 

12.85 

4,416.8 

520.37 

I.OOO 

o.oo 

2.2641 

• 

520.37 

12.90 

4,575-7 

522.38 

I.OOO 

o.oo 

2.1854 

" 

522.38 

12.95 

4,740.5 

524.39 

I.OOO 

o.oo 

2.1095 

" 

524.39 

13.00 

4,911.0 

526.41 

I.OOO 

o.oo 

2.0362 

" 

526.41 

13-05 

5,087.8 

528.43 

I.OOO 

o.oo 

1.9655 

u 

528.43 

13.10 

5,270.9 

530.44 

I.OOO 

o.oo 

1.8972 

u 

430.44 

13-15 

5,460.6 

532-45 

I.OOO 

o.oo 

1.8313 

u 

532.45 

13.20 

5,657.0 

534-47 

I.OOO 

o.oo 

1.7677 

u 

534-47 

13.25 

5,858.5 

536.49 

I.OOO 

o.oo 

1.7061 

u 

536.49 

13-30 

6,071.6 

538.50 

I.OOO 

o.oo 

1.6470 

u 

538.50 

13-35 

6,290.1 

540.51 

I.OOO 

o.oo 

1.5898 

u 

540.51 

13.40 

6,516.5 

542.53 

.000 

o.oo 

1.5346 

u 

542.53 

13-45 

6,751.0 

544-55 

.000  ' 

o.oo 

1.4813 

u 

544-55 

13-50 

6,993-9 

546.57 

.000 

o.oo 

1.4298 

u 

546.57 

13-55 

7,245-5 

548.58 

.000 

o.oo 

1.3801 

u 

548.58 

13.60 

551-00 

.000 

o.oo 

1.3322 

u 

551.00 

13-65 

7,776.4 

553-01 

I.OOO 

o.oo 

1.2859 

u 

553-01 

13.70 

8,056.4 

555-03 

I.OOO 

o.oo 

1.2412 

u 

555.03 

13-75 

8,346.2 

557-05 

I.OOO 

o.oo 

1.1982 

u 

557.05 

13.80 

8,646.7 

559-o6 

I.OOO 

o.oo 

1.1565 

u 

559.06 

13-85 

8,957.8 

561.07 

I.OOO 

o.oo 

1.1164 

u 

561.07 

13.90 

9,280.3 

563-09 

I.OOO 

o.oo 

1.0776 

a 

563.09 

13-95 

9,614.1 

565-11 

I.OOO 

o.oo 

1.0165 

u 

565.11 

14.00 

9,960.2 

567.12 

I.OOO 

o.oo 

1.0040 

u 

567.12 

14.05 

10,318 

569.14 

I.OOO 

o.oo 

9.69i4X 

I0~5 

569.14 

14.10 

10,690 

571.15 

I.OOO 

o.oo 

9-3547 

u 

571.15 

14-15 

11,075 

573-i6 

I.OOO 

o.oo 

9.0296 

u 

573-16 

14.20 

n,473 

575-iS 

I.OOO 

o.oo 

8.7160 

u 

575.18 

14-25 

11,886 

577-20 

I.OOO 

o.oo 

8.4132 

u 

577.20 

14.30 

12,314 

579-21 

I.OOO 

o.oo 

8.I2IO 

u 

579-21 

14-35 

12,757 

581.22 

I.OOO 

o.oo 

7.8388 

u 

581.22 

14.40 

13,216 

583-24 

I.OOO 

o.oo 

7.5666 

u 

583.24 

14-45 

13,692 

585-26 

.000 

o.oo 

7.3037 

u 

585.26 

14.50 

14,184 

587-27 

.000 

o.oo 

7.0500 

u 

587.27 

14-55 

14,695 

589.29 

.000 

o.oo 

6.8050 

u 

289.29 

14.60 

15,224 

591.30 

.000 

o.oo 

6.5687 

u 

591.30 

14.65 

15,772 

593-32 

.000 

o.oo 

6.3405 

u 

593.32 

14.70 

i6,339 

595-34 

.000 

o.oo 

6.1203 

u 

595-34 

14-75 

16,927 

597-35 

.000 

o.oo 

5.9077 

u 

597-35 

14.80 

17,536 

599-37 

.000 

o.oo 

5-7024 

a 

599-37 

14.85 

18,167 

601.39 

.000 

o.oo 

5-5044 

u 

601.39 

14.90 

18,822 

603.40 

.000 

o.oo 

5-3130 

u 

603.40 

14-9$ 

19,498 

605.41 

I.OOO 

o.oo 

5.1286 

u 

605.41 

15.00 

20,200 

607.43 

I.OOO 

o.oo 

4.9504 

u 

607.43 

Note.  Negative  quantities  are  in  heavy  type. 

Examples,     sinh  (14.0  /45°)  =  cosh  (14.0  745°)  =  9960.2  A67 


sech  (14.0  /45°) 


cosech  (14.0  /45°) 
[38] 


TABLE  VI 
FUNCTIONS  OF  SEMI-IMAGINARIES.   /(p/45JO  =  r/jy.    CONTINUED 


Sinh  and  cosh 


Tanh  and  coth 


Sech  and  cosech 


15-05 

20,927 

609.44 

I.OOO 

o.oo 

4-7785  Xio~6    609.44 

15.10 

21,680 

611.46 

1.  000 

o.oo 

4.6120 

u 

611.46 

15-15 

22.460 

613.48 

I.OOO 

o.oo 

4-4523 

u 

613.48 

15.20 

23,269 

615.49 

I.OOO 

o.oo 

4.2980 

u 

615.49 

15-25 

24,106 

617-50 

I.OOO 

o.oo 

4.1482 

u 

617.50 

15-30 

24,973 

619.52 

I.OOO 

o.oo 

4.0040 

u 

619.52 

15-35 

25,873 

621.54 

I.OOO 

o.oo 

3.8651 

u 

621.54 

15.40 

26,802 

623.55 

I.OOO 

o.oo 

3-73Jo 

u 

623.55 

15-45 

27,768 

625-57 

I.OOO 

o.oo 

3.6012 

u 

625.57 

15-50 

28,765 

627-59 

I.OOO 

o.oo 

3.4760 

u 

627.59 

15-55 

29,803 

630.00 

I.OOO 

o.oo 

3-3554 

u 

630.00 

15.60 

30,872 

632.02 

I.OOO 

o.oo 

3-2390 

u 

632.02 

15-65 

31,987 

634.04 

I.OOO 

o.oo 

3.1263 

a 

634.04 

15-70 

33,i4o 

636.05 

I.OOO 

o.oo 

3.0170 

u 

636.05 

15-75 

34,33i 

638.06 

I.OOO 

0.00 

2.9129 

u 

638.06 

15-80 

35,569 

640.08 

I.OOO 

o.oo 

2.8110 

u 

640.08 

15.85 

36,846 

642.10 

I.OOO 

o.oo 

2.7140 

a 

642.10 

15.90 

38,i74 

644.11 

I.OOO 

o.oo 

2.6200 

u 

644.11 

15-95 

39,546 

646.12 

I.OOO 

o.oo 

2.5287 

u 

646.12 

16.00 

40,970 

648.14 

I.OOO 

o.oo 

2.4410 

u 

648.14 

16.05 

42,443 

650.16 

I.OOO 

0.00 

2.3561 

a 

650.16 

16.10 

43,97i 

652.17 

I.OOO 

o.oo 

2.2740 

a 

652.17 

16.15 

45,553 

654.18 

I.OOO 

o.oo 

2.1952 

u 

654.18 

16.20 

47,192 

656.20 

I.OOO 

o.oo 

2.1190 

u 

656.20 

16.25 

48,890 

658.22 

I.OOO 

o.oo 

2.0454 

a 

658.22 

16.30 

50,649 

660.23 

I.OOO 

o.oo 

1.9740 

a 

660.23 

16.35 

52,473 

662.24 

I.OOO 

.  o.oo 

1-9055 

u 

662.24 

16.40 

54,359 

664.26 

I.OOO 

o.oo 

1.8400 

u 

664.26 

16.45 

56,316 

666.28 

I.OOO 

o.oo 

J-7757 

a 

666.28 

16.50 

58,475 

668.29 

I.OOO 

o.oo 

1.7100 

u 

668.29 

16.55 

60,444 

670.31 

I.OOO 

o.oo 

1.6544 

u 

670.31 

16.60 

62,619 

672.32 

I.OOO 

o.oo 

1.5969 

u 

672.32 

16.65 

64,872 

674-34 

I.OOO 

o.oo 

1-5415 

u 

674.34 

16.70 

57,208 

676-35 

I.OOO 

o.oo 

1.4879 

u 

676.35 

16.75 

69,626 

678.36 

I.OOO 

o.oo 

1.4362 

u 

678.36 

16.80 

72,132 

680.38 

I.OOO 

o.oo 

1.3863 

u 

680.38 

16.85 

74,727 

682.40 

I.OOO 

o.oo 

1.3382 

u 

682.40 

16.90 

77,4i8 

684.41 

I.OOO 

o.oo 

1.2917 

u 

684.41 

16.95 

80,203 

686.43 

I.OOO 

o.oo 

1.2468 

u 

686.43 

17.00 

83,088 

688.45 

I.OOO 

o.oo 

1-2035 

u 

688.45 

17-05 

86,080 

690.47 

I.OOO 

o.oo 

1.1617 

u 

690.47 

17.10 

89,176 

692.48 

I.OOO 

o.oo 

1.1214 

u 

692.48 

I7-I5 

92,387 

694.49 

I.OOO 

o.oo 

1.0824 

u 

694.49 

17.20 

95,  711 

696.51 

I.OOO 

o.oo 

1.0448 

u 

696.51 

17-25 

99,149 

698-53 

I.OOO 

o.oo 

1.0086 

u 

698-53 

Note.  Negative  quantities  are  in  heavy  type. 

Examples,     sinh  (17.0  /45°)  =  cosh  (17.0  /45°)  =  83,088  /688 


=  83,088  /328C 

sech  (17.0  /45°)  =  cosech  (17.0  /45°)  =  1-2035  X  io~5  \688°.45'- 
[39] 


TABLE  VI 
FUNCTIONS  OF  SEMI-IMAGINARIES. 

p  Sinh  and  cosh          Tanh  and  coth 


=  r/jy.    CONTINUED 


Sech  and  cosech 


17.30 

102,720 

700.54 

I.OOO 

o.oo 

9.7349X10-' 

'  700.54 

17-35 

106,420 

702.55 

I.OOO 

o.oo 

9.3968 

u 

702.55 

17.40 

110,250 

704.57 

I.OOO 

o.oo 

9.0703 

u 

704.57 

17.45 

114,220 

706.59 

I.OOO 

o.oo 

8.7551 

u 

706.59 

17-50 

"8,330 

709.00 

I.OOO 

o.oo 

8.4510 

u 

709.00 

17-55 

122,590 

711.01 

I.OOO 

o.oo 

8.1576 

u 

711.01 

17.60 

127,000 

7I3-03 

I.OOO 

o.oo 

7.8741 

u 

713.03 

17.65 

131,570 

715-05 

I.OOO 

o.oo 

7.6006 

u 

715.05 

17.70 

136,300 

717.06 

I.OOO 

o.oo 

7.3365 

u 

717.06 

17.75 

141,210 

719.07 

I.OOO 

o.oo 

7.0817 

u 

719.07 

17.80 

146,290 

721.09 

I.OOO 

o.oo 

6.8356 

u 

721.09 

17-85 

i5i,55o 

723.11 

I.OOO 

o.oo 

6.5983 

u 

723.11 

17.90 

157,000 

725.12 

I.OOO 

o.oo 

6.3710 

u 

725.12 

17-95 

162,660 

727.13 

I.OOO 

o.oo 

6.1478 

u 

727.13 

18.00 

168,520 

729-15 

I.OOO 

o.oo 

5.9383 

u 

729.15 

18.05 

174,580 

73I-I7 

I.OOO 

o.oo 

5.7281 

u 

731-17 

18.10 

180,860 

733-iS 

I.OOO 

o.oo 

5.5292 

u 

733.18 

18.15 

183,530 

735-20 

I.OOO 

o.oo 

5.4488 

u 

735-20 

18.20 

194,110 

737-21 

I.OOO 

o.oo 

5.1517 

u 

737.21 

18.25 

201,100 

739-23 

I.OOO 

o.oo 

4.9727 

u 

739.23 

18.30 

208,330 

741.24 

I.OOO 

o.oo 

4.8000 

u 

741.24 

18.35 

215,830 

743.26 

I.OOO 

o.oo 

4.6332 

u 

743.26 

18.40 

223,600 

745-27 

I.OOO 

o.oo 

4-4723 

u 

745.27 

18.45 

231,650 

747.29 

I.OOO 

o.oo 

4.3168 

u 

747.29 

18.50 

239,980 

749-31 

I.OOO 

o.oo 

4.1671 

u 

749.31 

18.55 

248,620 

75I-32 

I.OOO 

o.oo 

4.0222 

It 

751.32 

18.60 

257,570 

753-34 

I.OOO 

o.oo 

3.8825 

u 

753.34 

18.65 

266,840 

755-35 

I.OOO 

o.oo 

3.7476 

u 

755-35 

18.70 

.  276,440 

757-37 

I.OOO 

o.oo 

3.6174 

u 

757-37 

18-75 

286,390 

759-38 

I.OOO 

o.oo 

3.4918 

u 

759-38 

18.80 

296,690 

761.40 

I.OOO 

o.oo 

3.3628 

u 

761.40 

18.85 

307,380 

763-41 

I.OOO 

o.oo 

3.2533 

u 

763-41 

18.90 

318,570 

765-43 

I.OOO 

o.oo 

3.1404 

u 

765-43 

18.95 

329,890 

767.44 

I.OOO 

o.oo 

3.0313 

u 

767.44 

19.00 

34i,77o 

769.46 

I.OOO 

o.oo 

2.9260 

u 

769.46 

19.05 

354,060 

771-47 

I.OOO 

o.oo 

2.8244 

u 

771.47 

19.10 

366,810 

773-49 

I.OOO 

o.oo 

2.7262 

u 

773-49 

i9-!5 

380,010 

775-50 

I.OOO 

o.oo 

2.6315 

a 

775-50 

19.20 

393,690 

777-52 

I.OOO 

o.oo 

2.5401 

u 

777-52 

19-25 

407,850 

779-53 

I.OOO 

o.oo 

2.4519 

u 

779-53 

19.30 

422,530 

78i.55 

I.OOO 

o.oo 

2.3667 

u 

781.55 

19-35 

437,730 

783-57 

I.OOO 

o.oo 

2.2845 

u 

783.57 

19.40 

453,490 

785-59 

I.OOO 

o.oo 

2.2051 

u 

785.59 

19-45 

469,810 

788.00 

I.OOO 

o.oo 

2.1285 

u 

788.00 

19.50 

486,720 

790.02 

I.OOO 

o.oo 

2.0546 

u 

790.02 

Note.  Negative  qyantities  are  in  heavy  type. 

Examples,     sinh  (19.05  /45°)  =  cosh  (19.05  /45°)  =  354,o6o  /77i°- 
sech  (19.30  /45°)  =  cosech  (19.3  745°)  =  2.3667  X  io~«' 

[40] 


TABLE  VI 
FUNCTIONS  OF  SEMI-IMAGINARIES.   /(p/45°)  =  r  /y.    CONTINUED 


Sinh  and  cosh 


Tanh  and  coth 


Sech  and  cosech 


19-55 

504,230 

792.03 

I.OOO 

o.oo 

1.  983  2  > 

Cio~* 

1  792.03 

19.60 

522,380 

794-05 

I.OOO 

o.oo 

1-9*53 

u 

794.05 

19.65 

541,220 

796.06 

I.OOO 

o.oo 

1.8478 

" 

796.06 

19.70 

560,650 

798.08 

I.OOO 

o.oo 

1.7837 

" 

798.08 

19-75 

599,830 

800.09 

I.OOO 

o.oo 

1.6671 

" 

800.09 

19.80 

601,730 

802.  ii 

I.OOO 

o.oo 

1.6619 

« 

802.  II 

19.85 

623,390 

804.12 

I.OOO 

o.oo 

1.6041 

• 

804.12 

19.90 

645,820 

806.14 

I.OOO 

o.oo 

1.5484 

u 

806.14 

19-95 

669,070 

808.15 

I.OOO 

o.oo 

1.4946 

u 

808.15 

20.00 

693,1  50 

810.17 

I.OOO 

o.oo 

1.4426 

u 

810.17 

20.05 

718,090 

812.18 

I.OOO 

o.oo 

1.3926 

u 

812.18 

20.  10 

743,930 

814.20 

I.OOO 

o.oo 

1.3442 

u 

814.20 

20.15 

770,710 

816.21 

I.OOO 

o.oo 

1.2975 

u 

816.21 

20.20 

798,440 

818.23 

I.OOO 

o.oo 

1-2525 

u 

818.23 

20.25 

827,160 

820.24 

I.OOO 

o.oo 

1.2090 

u 

820.24 

20.30 

856,940 

822.26 

I.OOO 

o.oo 

1.1669 

u 

822.26 

20.35 

887,770 

824.27 

I.OOO 

o.oo 

1.1264 

a 

824.27 

20.40 

9*9,730 

826.29 

I.OOO 

o.oo 

1.0873 

u 

826.29 

20.45 

952,820 

828.30 

I.OOO 

o.oo 

1.0496 

u 

828.30 

20.50 

987,120 

830.32 

I.OOO 

o.oo 

1.0130 

u 

830.32 

Note.  Negative  quantities  are  in  heavy  type. 

Example,     sinh  (20.0  /45°)  =  cosh  (20.0  /45°)  =  693,i5o/8io°.i7/  =  693, 150 , 


[41] 


TABLE  VII.     HYPERBOLIC  SINES,     sinh  (x  +  iq)  =  u  +  iv 


x  =  0.05 


x  =  0.15 


0.0 

0.05 

O.I 

0.15 

O.2 

o.oo 
o.oo 
o.oo 
o.oo 
o.oo 

o.oo 
0.07846 

0.15643 
0-23345 

0.30902 

0.05002 
0.04987 

0.04945 

0.04864 

0.04757 

o.oo 
0.07856 
0.15663 
0.23374 
0.30940 

0.10017 
0.09986 
0.09893 
0.09740 
0.09526 

o.oo 
0.07885 
0.15722 
0.23461 
0.31056 

0.15056 

0.15010 
0.14871 
0.14640 

0.14319 

o.oo 

0.07934 

0.15820 
0.23608 
0.31250 

0.20134 

0.20072 

0.19886 
0.19577 

0.19148 

o.oo 
0.08003 

0.15957 
0.23813 
0.31522 

0.25 
0-3 
0-35 
0.4 

0-45 

o.oo 
o.oo 
o.oo 
o.oo 
o.oo 

0.38268 

Q-45399 
0.52250 

0.58778 
0.64944 

0.04621 

0.04457 

0.04265 
0.04047 
0.03804 

0.38316 
0-45454 
0.52313 
0.58850 
0.65023 

0.09254 
0.08925 
0.08541 
0.08104 
0.07617 

0.38460 
0.45626 
0.52511 

0.59073 
0.65270 

0.13910 

0.13415 
0.12838 

0.12181 
0.11449 

0.38700 
0.45911 
0.52839 
0.59441 
0.65677 

0.18601 
0.17939 
0.17167 
0.16288 
0.15310 

0.39036 
0.46310 

0.53298 
0.59958 

0.66248 

0-5 
0.55 

0.6 
0.65 
0.7 

o.oo 
o.oo 
o.oo 
o.oo 
o.oo 

0.70711 
0.76041 
0.80902 
0.85264 
0.89101 

0.03537 
0.03249 
0.02940 
0.02614 
0.02271 

0.70796 

0.76133 
0.81000 
0.85367 
0.89208 

0.07083 
0.06505 
0.05888 
0.05234 
0.04547 

0.71065 
0.76421 
0.81307 
0.85691 
0.89547 

0.10646 
0.09778 
0.08850 
0.07867 
0.06835 

0.71508 
0.76898 
0.81814 

0.86225 
0.90105 

0.14237 
0.13076 
0.11834 
0.10520 
0.09141 

0.72130 
0.77567 
0.82525 
0.86975 

0.90889 

0-75 
0.8 
0.85 
0.9 
0-95 

o.oo 
o.oo 
o.oo 
o.oo 
o.oo 

0.92388 

0.95106 

0.97237 

0.98769 
0.99692 

0.01914 
0.01546 
0.01168 
0.00783 
0.00392 

0.92503 

0.95225 

0-97359 
0.98892 
0.99816 

0.03833 
0.03095 
0.02338 
0.01567 
0.00786 

0.92850 
0.95582 
0.97724 
0.99263 
1.00191 

0.05762 
0.04653 
0.03515 
0.02355 

0.01181 

0.93429 

0.96178 

0.98333 

0.99882 
1.00815 

0.07705 
0.06222 
0.04700 
0.03150 
0.01580 

0.94242 
0.97014 
0.99188 
1.00751 
1.01692 

I.O 

1.05 
i.i 
I-I5 

1.2 

o.oo 
o.oo 
o.oo 
o.oo 
o.oo 

I.OOOOO 

0.99692 
0.98769 

0.97237 

0.95106 

o.oo 
0.00392 
0.00783 
0.01168 
0.01546 

1.00125 
0.99816 
0.98892 

0-97359 
0.95225 

o.oo 
0.00786 
0.01567 
0.02338 
0.03095 

1.00500 
1.00191 
0.99263 
0.97724 
0.95582 

o.oo 
0.01181 

0.02355 
0.03515 

0.04653 

1.01127 
1.00815 
0.99882 

0.98333 

0.96178 

o.oo 
0.01580 
0.03150 
0.04700 
0.06222 

1.02007 
1.01692 
1.00751 
0.99188 
0.97014 

I.2S 

i-3 
1-35 
1.4 

1-45 

o.oo 
o.oo 
o.oo 
o.oo 
o.oo 

0.92388 

0.89101 
0.85264 
0.80902 
0.76041 

0.01914 
0.02271 
0.02614 
0.02940 
0.03249 

0.92503 
0.89208 
0.85367 
0.81000 
0.76133 

0.03833 
0.04547 
0.05234 
0.05888 
0.06505 

0.92850 
0.89547 
0.85691 
0.81307 
0.76421 

0.05762 
0.06835 
0.07867 
0.08850 
0.09778 

0.93429 

0.90105 
0.86225 
0.81814 
0.76898 

0.07705 
0.09141 
0.10520 
0.11834 
0.13076 

0.94242 
0.90889 

0.86975 
0.82525 
0.77567 

i-5 
i-55 
1.6 
1.65 
i-7 

o.oo 
o.oo 
o.oo 
o.oo 
o.oo 

0.70711 

0.64944 

0.58778 

0.52250 
0-45399 

0.03537 
0.03804 
0.04047 
0.04265 

0.04457 

0.70796 
0-65023 
0.58850 
0.52313 
0-45454 

0.07083 
0.07617 
0.08104 
0.08541 
0.08925 

0.71065 
0.65270 
0-59073 
0-52511 
0.45626 

0.10646 
0.11449 
o.  12181 
0.12838 
0.13415 

0.71508 

0.65677 
0.59441 
0.52839 
0.45911 

0.14237 
0.15310 
0.16288 
0.17167 

0.17939 

0.72130 

0.66248 

0.59958 
0.53298 

0.46310 

1-75 
i.  80 

1.85 
1.9 
1-95 

o.oo 
o.oo 
o.oo 
o.oo 
o.oo 

0.38268 

0.30902 

0.23345 
0.15643 
0.07845 

0.04621 
0.04757 

0.04864 

0.04945 
0.04987 

0.38316 
0.30940 
0-23374 
0.15663 
0.07856 

0.09254 
0.09526 
0.09740 
0.09893 
0.09986 

0.38460 
0.31056 
0.23461 
0.15722 
0.07885 

0.13910 

0.14319 

0.14640 
0.14871 
0.15010 

0.38700 

0.31250 

0.23608 
0.15820 

0.07934 

o.  18601 
0.19148 

0.19577 
0.19886 
0.20072 

0.39036 
0.31522 
0.23813 
0.15957 

0.08003 

2.0 

o.oo 

o.oo 

0.05002 

o.oo 

0.10017 

o.oo 

0.15056 

o.oo 

0.20134 

o.oo 

Note. 

Negative  quantities  are  in  heavy  type. 

Examples. 

sinh  (o.i 

+  i  0.5)  =  0.07083 

+  *  0.71065. 

sinh  (o.i  +  i  1.2)  =  —0.03095  +  i  0.95582. 


[42] 


TABLE  VII.    HYPERBOLIC  SINES,     sinh  (x  +  iq)  =  u  +  iv.     CONTINUED 


x  =  0.25 


x  =  0.3 


=  0.35 


x  =  0.4 


x  =  0.45 


o 

0.25261 

o.oo 

0.30452 

o.oo 

0.35719 

o.oo 

0.41075 

o.oo 

0.46534 

o.oo 

0.05 

0.25183 

0.08092 

0.30358 

0.08202 

0.35609 

0.08331 

0.40949 

0.08482 

0.46391 

0.08654 

O.I 

0.24950 

0.16135 

0.30077 

0.16353 

0.35279 

0.16611 

0.40570 

0.16912 

0.45961 

0.17254 

0.15 

0.24563 

0.24078 

0.29611 

0.24403 

0-34732 

0.24789 

0-39940 

0.25237 

0.45249 

0.25748 

O.2 

0.24025 

0.31872 

0.28962 

0.32303 

0-33971 

0.32814 

0.39065 

0.33407 

0.44257 

0.34084 

0.25 

0.23338 

0.39471 

0.28134 

0.40003 

0.33000 

0.40636 

0-37949 

0.41371 

0.42992 

0.42209 

0-3 

0.22508 

0.46825 

0.27133 

0-47457 

0.31826 

0.48208 

0.36598 

0.49080 

0.41462 

0.50074 

o-35 

0.21539 

0.53891 

0.25965 

0.54619 

0.30455 

0.55483 

0.35022 

0.56486 

0.39677 

0.57630 

0.4 

0.20437 

0.60625 

0.24636 

0.61444 

0.28897 

0.62416 

0.33231 

0.63544 

0.37647 

0.64831 

0-45 

0.19208 

0.66985 

0.23156 

0.67889 

0.27161 

0.68964 

0.31234 

0.70210 

0.35385 

0.71632 

o-5 

0.17862 

0.72932 

0.21533 

0.73917 

0.25257 

0.75086 

0.29045 

0.76443 

0.32905 

0.77992 

Q-55 

o.  1  6406 

0.78429 

0.19777 

0.79488 

0.23198 

0.80746 

0.26676 

0.82206 

0.30222 

0.83871 

0.6 

0.14848 

0.83443 

0.17899 

0.84570 

0.20995 

0.85908  • 

0.24143 

0.87461 

0.27352 

0.89232 

0.65 

0.13199 

0.87942 

0.15911 

0.89130 

0.18663 

0.90540 

0.21462 

0.92177 

0.24314 

0.94044 

0.7 

0.11468 

0.91900 

0.13825 

0.93140 

0.16216 

0.94614 

0.18648 

0.96324 

0.21126 

0.98275 

0-75 

0.09667 

0.95290 

0.11654 

0.96577 

0.13669 

0.98105 

0.15719 

0.99878 

0.17808 

1.01901 

0.8 

0.07801 

0.98093 

0.09410 

0.99418 

0.11038 

.00991 

0.12693 

1.02816 

0.14380 

1.04899 

0.85 

0.05897 

1.00292 

0.07109 

1.01646 

0.08338 

•03254 

0.09589 

1.05120 

0.10863 

1.07250 

0.9 

0.03952 

1.01871 

0.04764 

1.03247 

0.05588 

.04880 

0.06426 

1.06776 

0.07280 

1.08939 

o-95 

0.01982 

1.02823 

0.02389 

1.04212 

0.02803 

.05860 

0.03223 

1.07774* 

0.03651 

1.09957 

I.O 

o.oo 

1.03141 

o.oo 

1-04534 

0.00 

.06188 

o.oo 

1.08107 

o.oo 

1.10297 

1.05 

0.01982 

1.02823 

0.02389 

1.04212 

0.02803 

.05860 

0.03223 

1.07774 

0.03651 

1.09957 

i.i 

0.03952 

1.01871 

0.04764 

1.03247 

0.05588 

.04880 

0.06426 

1.06776 

0.07280 

1.08939 

i-i5 

0.05897 

1.00292 

0.07109 

1.01646 

0.08338 

•03254 

0.09589 

1.05120 

0.10863 

1.07250 

1.2 

0.07801 

0.98093 

0.09410 

0.99418 

0.11038 

.00991 

0.12693 

1.02816 

0.14380 

1.04899 

1-25 

0.09667 

0.95290 

0.11654 

0.96577 

0.13669 

0.98105 

0.15719 

0.99878 

0.17808 

I.OIOOI 

i-3 

0.11468 

0.91900 

0.13825 

0.93140 

0.16216 

0.94614 

0.18648 

0.96324 

0.21126 

0.98275 

i-3S 

0.13199 

0.87942 

0.15911 

0.89130 

0.18663 

0.90540 

0.21462 

0.92177 

0.24314 

0.94044 

i-4 

0.14848 

0.83443 

0.17899 

0.84570 

0.20995 

0.85908 

0.24143 

0.87461 

0.27352 

0.89232 

1-45 

0.16406 

0.78429 

0.19777 

0.79488 

0.23198 

0.80746 

0.26676 

0.82206 

0.30222 

0.83871 

i-5 

0.17862 

0.72932 

0.21533 

0.73917 

0.25257 

0.75086 

0.29045 

0.76443 

0.32905 

0.77992 

1-55 

0.19208 

0.66985 

0.23156 

0.67889 

0.27161 

0.68964 

0.31234 

0.70210 

0.35385 

0.71632 

1.6 

0.20437 

0.60625 

0.24636 

0.61444 

0.28897 

0.62416 

0.33231 

0.63544 

0.37647 

0.64831 

1-65 

0.21539 

0.53891 

0.25965 

0.54619 

0.30455 

0.55483 

0.35022 

0.56486 

0.39677 

0.57630 

i-7 

0.22508 

0.46825 

0.27133 

0-47457 

0.31826 

0.48208 

0.36598 

0.49080 

0.41462 

0.50074 

i-75 

0.23338 

0.39471 

0.28134 

0.40003 

0.33000 

0.40636 

0.37949 

0.41371 

0.42992 

0.42209 

1.8 

0.24025 

0.31872 

0.28962 

0.32303 

0.33071 

0.32814 

0.39065 

0.33407 

0.44257 

0.34084 

1-85 

0.24563 

0.24078 

0.29611 

0.24403 

0.34732 

0.24789 

0.39940 

0.25237 

0.45249 

0.25748 

1.9 

0.24950 

0.16135 

0.30077 

0.16353 

0.35279 

0.16611 

0.40570 

0.16912 

0.45961 

0.17254 

i-95 

0.25183 

0.08092 

0.30358 

0.08202 

0.35609 

0.08331 

0.40949 

0.08482 

0.46391 

0.08654 

2.0 

0.25261 

o.oo 

0.30452 

o.oo 

0.35719 

o.oo 

0.41075 

o.oo 

0.46534 

o.oo 

Note. 

Negative  quantities  are  in  heavy  type. 

Examples. 

sinh  (0.4  +  i  o)  = 

0.41075  + 

to. 

sinh  (0.4  +  *  i)  = 

o.  +  i  1.08107. 

[43] 


TABLE  VII.     HYPERBOLIC   SINES,     sinh  (x  +  iq)  =  u  +  iv.     CONTINUED 


x  =  0.5 


=  0.55 


x  =  0.6 


x  =  0.65 


x  =  0.7 


o 
0.05 

O.I 

0.15 

O.2 

0.52110 
0.51949 
0.51468 
0.50670 
0-49559 

o.oo 
0.08847  • 
0.17640 
0.26324 
0.34846 

0.57815  o.oo 

0.57637  0.09063 

0.57103  0.18070 
0.56218  0.26965 
0.54986  0.35695 

0.63665 
0.63469 
0.62882 
0.61906 
0.60549 

o.oo 
0.09301 

0.18545 
0.27674 
0.36633 

0.69675 

0.69460 

0.68817 
0.67750 

0.66265 

o.oo 

0.09563 

0.19066 

0.28452 
0.37663 

0.75858 
0.75625 
0.74925 
0.73763 

0.72146 

o.oo 
0.09848 

0.19635 
0.29301 

0.38787 

O.25 
0-3 
0-35 
0.40 

0-45 

0.48143 
0.46430 

0.44431 
0.42158 
0.39624 

0.43152 

0.51193 
0.58918 
0.66280 

0.73233 

0.53414 
0.51514 

0.49296 

0.46773 
0.43963 

0.44204 

0.52441 
0.60354 
0.67895 
0.75018 

0.58819 
0.56726 
0.54284 
0.51506 
0.48412 

0.45366 
0.53819 

0.61940 
0.69680 
0.76990 

0.64371 

0.62081 
0.59408 

0.56368 

0.52981 

0.46641 

0.55332 
0.63682 
0.71639 
0.79154 

0.70084 
0.67590 
0.64680 

0.61371 
0.57683 

0.48033 

0.56984 

0.65582 

0.73777 
0.81517 

0-5 
0-55 

0.6 
0.65 
0.7 

0.36847 
0.33842 
0.30629 
0.27227 
0.23657 

0-79735 
0.85745 
0.91227 
0.96146 
1.00472 

0.40882 

0.37548 
0.33983 

0.30208 
0.26248 

0.81678 
0.87835 
0.93450 

0.98489 
1.02920 

0.45018 
0.41347 
0.37422 
0.33265 
0.28904 

0.83825 

0.90144 

0.95906 
1.01078 
1.05626 

0.49268 

0.45250 
0.40954 
0.36405 
0.31632 

0.86182 
0.92678 
0.98602 

1.03919 
1.08595 

0.53640 

0.49266 

0.44589 
0.39636 
0.34439 

0.88754 
0.95444 
1.01545 

1.07021 

1.11836 

0-75 
0.8 
0.85 
0.9 
0-95 

0.19942 
0.16103 
0.12165 
0.08152 
0.04088 

1.04179 
1.07244 
1.09647 

I-II374 
1.12415 

0.22125 

0.17866 
0.13497 

0.09044 
0.04536 

1.06717 

1.09857 
1.12319 
1.14088 
1.15154 

0.24364 
0.19674 
0.14862 
0.09959 
0.04995 

1.09523 
1.12744 
1.15271 
1.17087 
1.  18181 

0.26663 

0.21531 

0.16265 
0.10900 

0.05467 

1.12602 
1.15912 
1.18512 
1.20379 
1.21504 

0.29030 

0.23442 
0.17709 

0.11867 

0.05952 

1.15962 
1.19374 

1.22049 

1.23972 
1.25130 

.0 

•05 
.1 

•15 

.2 

o.oo 
0.04088 
0.08152 
0.12165 
0.16103 

1.12763 
1.12415 

I-II374 
1.09647 
1.07244 

o.oo 
0.04536 
0.09044 

0.13497 
0.17866 

1.15510 
1.15154 
1.14088 
1.12319 
1.09857 

0.00 

0.04995 
0.09959 

0.14862 
0.19674 

1.18547 
1.  18181 

1.17087 

1.15271 
1.12744 

o.oo 
0.05467 
0.10900 
0.16265 

0.21531 

1.21879 
1.21504 
1.20379 
1.18512 
1.15912 

o.oo 
0.05952 
0.11867 

0.17709 
0.23442 

1.25517 
1.25130 
1.23972 

1.22049 

1-19374 

1-25 

i-35 
1.4 

i  -45 

0.19942 

0.23657 

0.27227 
0.30629 

0.33842 

1.04179 
1.00472 
0.96146 
0.91227 
0.85745 

0.22125 
0.26248 
0.30208 

0.33983 
0.37548 

1.06717 

1.02920 
0.98489 

0.93450 
0.87835 

0.24364 
0.28904 

0.33265 
0.37422 
0.41347 

1.09523 

1.05626 
1.01078 
0.95906 
0.90144 

0.26663 
0.31632 
0.36405 
0.40954 
0.45250 

1.12602 

1.08595 
1.03919 

0.98602 
0.92678 

0.29030 

0.34439 
0.39636 
0.44589 

0.49266 

1.15962 
1.11836 

1.07021 

1-01545 

0-95444 

i-SS 
1.6 
1-65 

0.36847 

0.39624 
0.42158 

0.44431 

0.46430 

0-79735 
0.73233 
0.66280 
0.58918 
0.5H93 

0.40882 

0.43963 
0.46773 

0.49296 

0.51514 

0.81678 
0.75018 
0.67895 
0.60354 
0.52441 

0.45018 
0.48412 
0.51506 
0.54284 
0.56726 

0.83825 

0.76990 
0.69680 
0.61940 

0.53819 

0.49268 
0.52981 
0.56368 
0.59408 
0.62081 

0.86182 
0.79154 
0.71639 
0.63682 
0.55332 

0.53640 
0.57683 

0.61371 
0.64680 
0.67590 

0.88754 
0.81517 

0-73777 
0.65582 
0.56984 

1-75 
1.8 
1.85 
1.9 
1-95 

0.48143 

0.49559 

0.50670 
0.51468 

0.51949 

0.43152 
0.34846 
0.26324 
o.  1  7640 
6.08847 

0.53414 

0.54986 
0.56218 
0.57103 
0.57637 

0.44204 

0.35695 

0.26965 
0.18070 
0.09063 

0.58819 
0.60549 
0.61906 
0.62882 
0.63469 

0.45366 
0.36633 
0.27674 
0.18545 

0.09301 

0.64371 
0.66265 
0.67750 
0.68817 

0.69460 

0.46641 

0.37663 
0.28452 

0.19066 

0.09563 

0.70084 
0.72146 

0.73763 
0.74925 
0.75625 

0.48033 
0.38787 
0.29301 
0.19635 
0.09848 

2.0 

0.52110 

0.00 

0.57815 

0.00 

0.63665 

o.oo 

0.69675 

0.00 

0.75858 

o.oo 

Note. 

Negative  quantities  are  in  heavy  type. 

Examples. 

sinh  (0.65 
sinh  (0.55  • 

f  i  0-75) 
f  i  1-40) 

=  0.26663  +  *  1.12602. 

=  -0.33983  +  *  0.93450. 

[44] 


TABLE  VII.    HYPERBOLIC  SINES,     sinh  (x  +  iq)  =  u  +  iv.     CONTINUED 


x  =  0.75 


x  =  0.85 


x  =  o.g 


x  =  0.95 


o 

0.05 

O.I 

0-15 

O.2 

0.82232 
0.81978 
0.8I2I9 
0.79960 
0.78207 

o.oo 
0.10158 

0.20253 

0.30224 
0.40079 

0.88811 

0.88537 

0.87717 
0.86357 

0.84464 

o.oo 
0.10493 
0.20922 
0.31222 
0.41329 

0.95612 
0.95317 
0-94435 
0.92970 
0.90932 

o.oo 

0.10855 

0.21643 
0.32298 
0.42753 

1.02652  . 

1.02335 
1.01388 

0.99816 
0.97628 

o.oo 

0.11244 
0.22418 

0.33455 
0.44285 

1  .09948 
1.09610 

1.08595 

1.06911 

1.04567 

o.oo 
o.  11661 

0.23250 

0.34695 
0.45927 

O.25 

o-3 

0.35 
0.4 

0.45 

0.75972 
0.73269 
O./OII4 
0.66527 
0.62529 

0.49545 
0.58777 

0.67647 
0.76100 
0.84083 

0.82050 

0.79131 
0.75724 

0.71849 

0.67532 

0.51182 
0.60718 
0.69881 
0.78613 
0.86859 

0.88334 
0.85191 
0.81522 

o.7735i 
0.72704 

0.52945 
0.62811 
0.72289 
0.81322 

0.89853 

0.94838 
0.91463 
0.87525 
0.83047 
0.78057 

0.54842 

0.65061 

0.74879 
0.84235 
0.93071 

1.01579 
0.97965 
0.93747 

0.88950 

0.83605 

0.56875 
0.67473 
0.77655 
0.87358 

0.96523 

°-5 
0-55 
0.6 
0.65 

0.7 

0.58146 
0.53405 
0.48335 
0.42966 

0-37332 

0.91548 
0.98449 
1.04742 
1.10390 
I-iS3S6 

0.62799 

0.57678 

0.52202 

0.46403 
0.40319 

0.94571 
1.01700 
1.08201 

1-14035 
1.19166 

0.67608 
0.62095 
0.56199 

0-49957 
0.43407 

0.97830 
1.05205 
1.11930 
1.17965 
1.23274 

0.72586 
0.66667 
0.60337 
0.53635 

0.46603 

1-01334 
1.08973 

i-!5939 
1.22191 
1.27689 

0.77745 

0.71406 
0.64621 

0.57448 

0.49916 

1.05092 
1.13013 
1.20238 
1.26722 

1-32425 

0.75 
0.8 
0.85 
0.9 
o-95 

0.31469 
0.254II 
0.19197 
0.12864 
0.06452 

1.19613 
1.23132 
1.25891 
1.27874 
1.29069 

0.33986 
0.27444 
0.20732 
0.13893 

0.06968 

1-23563 
1.27198 
1.30048 
1.32097 
J-3333i 

0.36589 
0.29546 
0.22320 
0.14957 
0.07502 

1.27822 

1.31582 
1.34530 
1.36650 
1.37927 

0.39283 
0.31721 
0.23964 

0.16058 
0.08054 

1.32400 
1.36294 

1-39349 
1.41544 
1.42867 

0.42076 

0.33976 

0.25667 
0.17200 
0.08627 

1-37309 
1.41348 
1.44516 
1.46793 

1.48164 

I.O 

1.05 
i.i 
1-15 

1.2 

o.oo 
0.06452 
0.12864 
0.19197 
0.25411 

1.29468 
1.29069 
1.27874 
1.25891 
1.23132 

o.oo 
0.06968 

0.13893 
0.20732 
0.27444 

1-33743 
•33331 
•32097 
.30048 
.27198 

o.oo 
0.07502 

0.14957 

0.22320 
0.29546 

1.38353 
•37927 
•36650 
•34530 

.31582 

o.oo 
0.08054 
0.16058 
0.23964 
0.31721 

I.43309 
1.42867 
1.41544 

1-39349 
1.36294 

o.oo 
0.08627 
0.17200 
0.25667 

0.33976 

1.48623 

1.48164 

1.46793 
1.44516 
1.41348 

1-25 

i-3 
i-3S 
1.4 

1-45 

0.31469 
0.37332 

0.42966 

0.48335 
0.53405 

1.19613 
I-I5356 
1.10390 
1.04742 
0.98449 

0.33986 
0.40319 
0.46403 
0.52202 

0.57678 

•23563 
.19166 

•14035 
.08201 
.01700 

0.36589 
0.43407 
0.49957 
0.56199 
0.62095 

.27822 

.23274 

•17965 
.11930 
.05205 

0.39283 

0.46603 

0.53635 

0.60337 
0.66667 

1.32400 
1.27689 
1.22191 

I-I5939 
1.08973 

0.42076 
0.49916 

0.57448 

0.64621 
0.71406 

1.37309 
1.32425 

1.26722 
1.20238 
1.13013 

i-5 
i-SS 
1.6 

1.65 

i-7 

0.58146 
0.62529 
0.66527 
0.70114 
0.73269 

0.91548 
0.84083 
0.76100 
0.67647 
0.58777 

0.62799 

0.67532 

0.71849 

0.75724 
0.79131 

0.94571 
0.86859 
0.78613 
0.69881 
0.60718 

0.67608 
0.72704 
0.77351 
0.81522 
0.85191 

0.97830 
0.89853 

0.81322 
0.72289 
0.62811 

0.72586 
0.78057 
0.83047 

0.87525 

0.91463 

i-oi334 
0.93071 
0.84235 
0.74879 
0.65061 

0.77745 
0.83605 
0.88950 
0-93747 
0.97965 

1.05092 
0.96523 
0.87358 
0.77655 
0.67473 

i-75 
1.8 
1.85 
1.9 

i-95 

0.75972 

0.78207 
0.79960 
0.81219 
0.81978 

0-49545 
0.40079 
0.30224 
0.20253 
0.10158 

0.82050 
0.84464 

0.86357 
0.87717 
0.88537 

0.51182 
0.41329 
0.31222 
0.20922 
0.10493 

0.88334 
0.90932 
0.92970 
0.94435 
0.95317 

0.52945 
0.42753 
0.32298 
0.21643 
0.10855 

0.94838 
0.97628 
0.99816 
1.01388 
1-02335 

0.54842 
0.44285 

0-33455 
0.22418 
0.11244 

1.01579 
1.04567 
1.06911 
1.08595 
1.09610 

0.56875 
0.45927 

0-34695 
0.23250 
0.11661 

2.0 

0.82232 

o.oo 

0.88811 

o.oo 

0.95612 

o.oo 

1.02652 

o.oo 

1.09948 

o.oo 

Note. 

Negative  quantities  are  in  heavy  type. 

Examples. 

sinh  (0.8  +  *  0.7) 

=  0.40319  -f-  *  1.19166. 

sinh  (0.8  +  i  1.7)  =  —0.79131  +  10.60718. 


[45] 


TABLE  VII.     HYPERBOLIC  SINES,     sinh  (*  +  iq)  =  u  +  iv.     CONTINUED 


=  1.05 


=  1.15 


0 

0.05 

O.I 

0.15 

0.2 

1.17520  o.oo 

1.17158  O.I2I07 
I.I6073  0.24139 
1.14273  0.36023 
I.II768  0.47684 

1.25386  o.oo 

1.24999  0.12583 
1.23842  0.25089 
1.21921  0.37440 
1.19249  0.49560 

1-33565  o.oo 

1-33153  0.13091 

1.31920  0.26101 

1.29875  0.38951 

1.27028  0.51560 

1.42078  o.oo 
1.41640  0.13632 
1.40329  0.27179 
1.38152  0.40559 
1.35124  0.53689 

1.50946  o.oo 
1.50481  0.14206 
1.49088  0.28325 
1.46776  0.42269 

1.43558  0.55952 

0.25 

o-3 
0-35 
0.4 

0-45 

1.08574  0.59051 
I.047II  0.70055 
I.OO2O2  0.80626 
0.95076  0.90700 
0.89363   I.002I5 

1.15841  0.61375 

1.11719  0.72811 
1.06909  0.83798 
1.01439  0.94269 

0.95344  1.04158 

1.23398  0.63852 
1.19007  0.75749 
1.13883  0.87180 

1.08056  0.98073 
1.01564  1.08362 

1.31263  0.66488 
1.26592  0.78877 
1.21141  0.90780 

1.14943  1.02123 

1.08037  1.12836 

1.39456  0.69291 

1.34494  0.82202 
1.28703  0.94607 
1.  22118  1.06428 

1.14781  1.71593 

o-5 
0-55 
0.6 
0.65 
0.7 

0.83099  I.O9II2 
0.76323  I.I7337* 
0.69077  1.24838 
0.61404  1.31569 

0-53353  1-3749° 

0.88661  1.13405 
0.81432  1.21953 
0.73700  1.29750 
0.65514  1.36746 

0.56924  1.42899 

0.94445  1.17982 
0.86743  1.26875 
0.78508  1.34986 

0.69787  1.42265 
0.60637  1.48666 

1.00464  1.22854 
0.92272  1.32114 
0.83511  1.40560 
0.74235  1.48139 
0.64502  1.54805 

1.06735  1-28033 
0.98032  1.37684 
0.88724  1.46485 

0.78869  1.54384 
0.68528  1.61331 

0-75 
0.8 
0.85 
0.9 
0-95 

0.44973  1.42562 
0.36316  1.46756 
0.27435  I-50045 
0.18384  1.52408 
0.09221  1.53832 

0.47983  1.48171 
0.38746  1.52530 
0.29271  1.55948 
0.10615  1.58405 
0.09838  1.59885 

0.51113  1.54151 

0.41274  1.58685 
0.31180  1.62242 
0.20894  1.64798 
0.10479  1.66337 

0.54371  1.60517 
0.43904  1.65238 
0.33167  1.68941 

0.22226  1.71602 
0.11147  1.73206 

0.57765  1.67283 

0.46645  1.72204 

0.35238  1.76063 
0.23613  1.78836 
0.11843  1.80507 

I.O 

1-05 
i.i 
i.iS 

1.2 

o.oo    1.54308 

0.09221  1.53832 

0.18384  1.52408 

0.27435  1.50045 
0.36316  1.46756 

o.oo    1.60379 
0.09838  1.59885 
0.19615  1.58405 
0.29271  1.55948 
0.38746  1.52530 

o.oo  1.66852 

0.10479  1.66337 

0.20894  1.64798 
0.31180  1.62242 

0.41274  1.58685 

0.00     I.7374I 

0.11147  1.73206 
0.22226  1.71602 
0.33167  1.68941 

0.43904  1.65238 

o.oo    1.81066 
0.11843  1.80507 
0.23613  1.78836 
0.35238  1.76063 

0.46645  1.72204 

1.25 

i-3 
i-35 
1.4 

i-45 

0.44973  1.42562 
0.53353  1.37490 

0.61404  1.31569 
0.69077  1.24838 

0.76323  1.17337 

0.47983  1.48171 
0.56924  1.42899 
0.65514  1.36746 
0.73700  1.29750 

0.81432  1.21953 

0.51113  1.54151 

0.60637  1.48666 
0.69787  1.42265 
0.78508  1.34986 
0.86743  1.26875 

0.54371  1.60517 
0.64502  1.54805 
0.74235  1.48139 

0.83511   .40560 
0.92272   .32114 

0.57765  1.67283 
0.68528  1.61331 

0.78869  1.54384 

0.88724  1.46485 
0.98032  1.37684 

i-5 
i-SS 
1.6 
1.65 
i-7 

0.83099  1.09112 
0.89363  1.00215 
0.95076  0.90700 

I.OO2O2  0.80626 
I.047II  0.70055 

0.88661  1.13405 

0.95344  1.04158 

.01439  0.94269 
.06909  0.83798 
.11719  0.72811 

0.94445  1.17982 
1.01564  1.08362 
1.08056  0.98073 
1.13883  0.87180 
1.19007  0.75749 

1.00464   .22854 
1.08037  .12836 
1.14943  .02123 
1.21141  0.90780 
1.26592  0.78877 

1.06735  1.28033 
1.14781  1.17593 
1.22118  1.06428 
1.28703  0.94607 
1.34494  0.82202 

i-7S 
1.8 
1.85 
1.9 
i-9S 

1.08574  0.59051 
I.II768  0.47684 
I.I4273  0.36023 
I.I6073  0.24139 
I.I7I58  O.I2IO7 

.15841  0.61375 
.19249  0.49560 
.21921  0.37440 
.23842  0.25089 

.24999  0.12583 

1.23398  0.63852 

1.27028  0.51560 

1.29875  0.38951 

1.31920  0.26101 

1.33153  0.13091 

1.31263  0.66488 

1.35124  0.53689 
1.38152  0.40559 
1.40329  0.27179 

1.41640  0.13632 

1.39456  0.69291 
1-43558  0.55952 

1.46776  0.42269 
1.49088  0.28325 
1.50481  0.14206 

2.O 

1.17520  o.oo 

1.25386  o.oo 

1.33565  o.oo 

1.42078  o.oo 

1.50946  o.oo 

Note.  Negative  quantities  are  in  heavy  type. 

Examples,     sinh  (i.o  +  *  i.o)  =  o  +  i  1.54308. 

sinh  (i.o  +  i  1.5)  =  —0.83099  +  i  1.09112. 


[46] 


TABLE  VII.    HYPERBOLIC  SINES,     sinh  (x  +  iq)  =  u  +  iv.     CONTINUED 


=  1.25 


x  =  1.3 


*  =  i-35 


*  =  1.4 


=  1-45 


o 

1.60192  o.oo 

1.69838  o.oo 

1.79909  o.oo 

1.90430  o.oo 

2.01427  o.oo 

0.05 

1.59698  0.14816 

1.69315  0.15464 

1.79354  0.16150 

1.89843  0.16876' 

2.00806  0.17644 

O.I 

1.58220  0.29541 

1.67747  0.30832 

1.77694  0.32199 

1.88086  0.33647 

1.98947  0.35180 

0.15 

1.55766  0.44084 

1.65146  0.46010 

1.74938  0.48051 

1.85169  0.50212 

1.95862  0.52498 

O.2 

1.52352  0.58355 

1.61526  0.60905 

1.71104  0.63606 

1.81110  0.66466 

1.91569  0.69493 

0.25 

1.47998  0.72267 

1.56910  0.75424 

1.66215  0.78769 

1.75934  0.82311 

1.86094  0.86060 

o-3 

1.42732  0.85733 

1.51327  0.89478 

1.60300  0.93446 

1.69675  0.97649 

1.79473  1.02095 

o-35 

1.36586  0.98670 

1.44811  1.02980 

1.53398  1.07548 

1.62369  1.12384 

1.71745  1.17502 

0.4 

1.29598  1.10999 

1.37402  1.15848 

1.45550  1.20986 

1.54061  1.26427 

1.62958  1.32184 

0-45 

1.  21811  1.22643 

1.29146  1.28001 

1.36804  1.33678 

1.44804  1.39690 

1.53166  1.46051 

°-5 

1.13273  1.33532 

1.20094  1.39365 

1.27215  1.45546 

1.34655  1.52092 

1.42431  1.59017 

Q-55 

1.04036  1.43597 

1.10301  1.49870 

1.16842  1.56517 

1.23674  1.63556 

1.30817  1.71007 

0.6 

0.94158  1.52777 

0.99829  1.59451 

1.05748  1.66523 

1.11932  1.74012 

1.18396  1.81935 

0.65 

0.83700  1.61014 

0.88740  1.68048 

0.94002  1.75502 

0.99500  1.83394 

1.05245  1.91745 

0.7 

0.72726  1.68260 

0.77105  1.75610 

0.81677  1.83399 

0.86454  1.91646 

0.91446  2.00373 

0-75 

0.61303  1.74467 

0.64994  1.82089 

0.68848  1.90165 

0.72875  1.98717 

0.77083  2.07766 

0.8 

0.49502  1.79600 

0.52483  1.87445 

0-55595  1-95759 

0.58846  2.04562 

0.62244  2.13878 

0.85 

0.37396  1.83624 

0.39648  1.91646 

0.41999  2.00146 

0.44455  2.09147 

0.47022  2.18671 

0.9 

0.25060  1.86517 

0.26569  1.94665 

0.28144  2.03299 

0.29790  2.12442 

0.31510  2.22115 

o-9S 

0.12569  1.88260 

0.13325  1.96484 

0.14116  2.05199 

0.14427  2.14427 

0.15804  2.24191 

I.O 

o.oo    1.88842 

o.oo    1.97091 

o.oo    2.05833 

o.oo    2.15090 

o.oo    2.24884 

1.05 

0.12569  1.88260 

0.13325  1.96484 

0.14116  2.05199 

0.14427  2.14427 

0.15804  2.24191 

i.i 

0.25060  1.86517 

0.26569  1.94665 

0.28144  2.03299 

0.29790  2.12442 

0.31510  2.22115 

i-J5 

0.37396  1.83624 

0.39648  1.91646 

0.41999  2.00146 

0.44455  2.09147 

0.47022  2.18671 

1.2 

0.49502  1.79600 

0.52483  1.87445 

0-55595  1-95759 

0.58846  2.04562 

0.62244  2.13878 

I-25 

0.61303  1.74467 

0.64994  1-82089 

0.68848  1.90165 

0.72875  1-98717 

0.77083  2.07766 

i-3 

0.72726  1.68260 

0.77105  1.75610 

0.81677  1-83399 

0.86454  1.91646 

0.91446  2.00373 

i-35 

0.83700  1.61014 

0.88740  1.68048 

0.94002  1.75502 

0.99500  1.83394 

1.05245  1.91745 

1.4 

0.94158  1.52777 

0.99829  1.59451 

1.05748  1.66523 

1.11932  1.74012 

1.18396  1.81935 

i-45 

1.04036  1.43597 

1.10301  1.49870 

1.16842  1.56517 

1.23674  1.63556 

1.30817  1.71007 

i-5 

1.13273  1.33532 

.20094  1.39365 

1.27215  1.45546 

1-34655  1.52092 

1.42431  1.59017 

i-55 

1.  21811  1.22643 

.29146  1.28001 

1.36804  1.33678 

1.44804  1.39690 

1.53166  1.46051 

1.6 

1.29598  1.10999 

.37402  1.15848 

I-45550  1.20986 

1.54061  1.26427 

1.62958  1.32184 

1.65 

1.36586  0.98670 

.44811  1.02980 

I-53398  1.07548 

1.62369  1.12384 

1.71745  1.17502 

i-7 

1.42732  0.85733 

.51327  0.89478 

1.60300  0.93446 

1.69675  0.97649 

1-79473  1-02095 

i-75 

1.47998  0.72267 

.56910  0.75424 

1.66215  0.78769 

1-75934  0.82311 

1.86094  0.86060 

1.8 

1.52352  0.58355 

.61526  0.60905 

1.71104  0.63606 

1.81110  0.66466 

1.91569  0.69493 

1.85 

1-55766  0.44084 

.65146  0.46010 

1.74938  0.48051 

1.85169  0.50212 

1.95862  0.52498 

1.9 

1.58220  0.29541 

.67747  0.30832 

1.77694  0.32199 

1.88086  0.33647 

1.98947  0.35180 

i-95 

1.59698  0.14816 

.69315  0.15464 

1-79354  0.16150 

1.89843  0.16876 

2.00806  0.17644 

2.O  I.6OI92     O.OO 


1.69838   o.oo 


1.79909    o.oo 


1.90430   o.oo 


2.01427   o.oo 


Note.  Negative  quantities  are  in  heavy  type. 

Examples,     sinh  (1.35  +  to)  =  1.70909  +  io. 

sinh  (1.4  +  i  1.15)  =  -0.44455  +  i  2.09147. 


[47] 


TABLE  VII.    HYPERBOLIC  SINES,     sinh  (x  +  iq)  =  u  +  iv.     CONTINUED 


*  =  1.5 


=  i-SS 


1.6 


*  =  1.65 


x  =  1.7 


0 

0.05 

O.I 

0.15 

0.2 

2.12928  o.oo 
2.12272  0.18457 
2.10307  0.36800 
2.07045  0.54916 
2.02507  0.72693 

2.24961  o.oo 
2.24268  0.19316 
2.22191  0.38512 

2.18745  0.57471 
2.13951  0.76076 

2-37557  0.00 

2.36824  0.20223 
2.34632  0.40320 

2.30993  0.60170 

2.25930  0.79648 

2.50746  o.oo 
2.49973  0.21180 

2.47659  0.42230 

2.43818  0.63019 
2.38474  0.83420 

2.64563  o.oo 

2.63747  0.22191 

2.61306  0.44245 
2.57253  0.66026 
2.51614  0.87400 

0.25 

o-3 
o-3S 
0.4 

o-4S 

1.96720  0.90023 
1.89720  1.06797 
1.81551  1.22913 
1.72263  1.38271 
1.61912  1.52777 

2.07837  0.94211 
2.00442  1.11766 
1.91811  1.28632 

1.81997  1.44704 
1.71062  1.59885 

2.19473  0.98636 

2.11664  1.17015 
2.02550  1.34672 
1.92187  1.51500 
1.80640  1.67393 

2.31660  1.03306 

2.23417  1.22556 
2.13797  1.41050 

2.02858  1.58674 
1.90669  1.75320 

2.44424  1.08235 
2.35727  1.28403 

2.25577  1-47779 
2.14036  1.66244 
2.01175  1-83684 

o-S 
0-55 
0.6 
0.65 
0.7 

1  -5°563  1-66341 
1.38286  1.78879 
1.25156  1.90314 
1.11255  2.00576 
0.96667  2.09601 

1.59071  1.74080 

1.46101  1.87201 
1.32229  1.99169 
1.17542  2.09908 
1.02130  2.19353 

1.67978  1.82254 
1.54281  1.95992 
1.39632  2.08522 
1.24123  2.19765 
1.07848  2.29654 

1-77305  1.90885 
1.62847  2.05274 
1.47385  2.18396 
1.31015  2.30173 
1.13837  2.40529 

1.87074  1.99992 
1.71820  2.15067 
1.55506  2.28816 
1.38234  2.41154 
1.20109  2.52005 

0-75 
0.8 
0.85 
0.9 
o.QS 

0.81484  2.17334 
0.65798  2.23727 
0.49707  2.28742 
0.33309  2.32345 
0.16706  2.34516 

0.86089  2.27446 

0.69517  2.34137 
0.52516  2.39384 
0.35192  2.43155 
0.17650  2.45427 

0.90909  2.38127 

0.73409  2.45131 
0.55456  2.50625 
0.37163  2.54573 
0.18639  2.56952 

0.95957  2.49404 
0.77485  2.56740 
0.58536  2.62494 

0.39225  2.66629 
0.19673  2.69121 

1.01244  2.61302 
0.81754  2.68989 
0.61761  2.75017 
0.41387  2.79350 
0.20757  2.81960 

I.O 

1.05 
i.i 
i-iS 

1.2 

o.oo    2.35241 
0.16706  2.34516 

0.33309  2.32345 

0.49707  2.28742 

0.65798  2.23727 

o.oo    2.46186 

0.17650  2.45427 
0.35192  2.43155 
0.52516  2.39384 
0.69517  2.34137 

o.oo  2.57746 

0.18639  2.56952 
0.37163  2.54573 
0.55456  2.50625 
0.73409  2.45131 

o.oo  2.69951 
0.19673  2.69121 
0.39225  2.66629 
0.58536  2.62494 
0.77485  2.56740 

o.oo  2.82832 
0.20757  2.81960 

0.41387  2.79350 

0.61761  2.75017 
0.81754  2.68989 

1.25 

i-3 
i-35 
1.4 

1-45 

0.81484  2.17334 

0.96667  2.09601 
1.  11255  2.00576 
1.25156  1.90314 
1.38286  1.78879 

0.86089  2.27446 

1.02130  2.19353 

1.17542  2.09908 
1.32229  1.99169 
1.46101  1.87201 

0.90909  2.38127 
1.07848  2.29654 

1.24123  2.19765 
1.39632  2.08522 
1.54281  1.95992 

0.95957  2.49404 
1.13837  2.40529 
1.31015  2.30173 
1.47385  2.18396 
1.62847  2.05274 

1.01244  2.61302 
1.20109  2.52005 

1.38234  2.41154 
1.55506  2.28816 
1.71820  2.15067 

i-5 
i-5S 
1.6 
1.65 
i-7 

1.50563  1.66341 
1.61912  1.52777 
1.72263  1.38271 

1.81551  1.22913 
1.89720  1.06797 

1.59071  1.74080 
1.71062  1.59885 
1.81997  1.44704 
1.91811  1.28632 
2.00442  1.11766 

1.67978  1.82254 
1.80640  1.67393 
1.92187  1.51500 
2.02550  1.34672 

2.11664  1.17015 

1.77305  1.90885 
1.90669  1.75320 
2.02858  1.58674 

2.13797  1.41050 
2.23417  1.22556 

1.87074  1.99992 
2.01175  1.83684 
2.14036  1.66244 
2.25577  1-47779 
2.35727  1-28403 

1-75 
1.8 
i.8S 
1.9 

i-95 

1.96720  0.90023 
2.02507  0.72693 
2.07045  0.54916 
2.10307  0.36800 
2.12272  0.18457 

2.07837  0.94211 
2.13951  0.76076 

2.18745  0.57471 

2.22191  0.38512 
2.24268  0.19316 

2.19473  0.98636 

2.25930  0.79648 
2.30993  0.60170 
2.34632  0.40320 
2.36824  0.20223 

2.31660  1.03306 
2.38474  0.83420 

2.43818  0.63019 

2.47659  0.42230 

2.49973  0.21  180 

2.44424  1.08235 
2.51614  0.87400 
2.57253  0.66026 
2.61306  0.44245 
2.63747  0.22191 

2.0 

2.12928  o.oo 

2.24961  o.oo 

2-37557  o.oo 

2.50746  o.oo 

2.64563  o.oo 

Note.     Negative  quantities  are  in  heavy  type. 

Examples,  sinh  (1.7  -f  1*0.7)  =  1.20109  4 

•  /  2.52005. 

sinh  (1.7  +  11-2)  =  -2.35727  +  *  1.28403. 


[48] 


TABLE  VII.    HYPERBOLIC  SINES,     sinh  (x  +  iq)  =  u  +  iv,    CONTINUED 


=  i-7S 


x  =  1.8 


=  1.85 


*  =  1.9 


x  =  1.95 


o 

2.79041 

o.oo 

2.94217 

o.oo 

3.10129 

o.oo 

3.26816 

o.oo 

3.44321 

o.oo 

0.05 

2.78181 

0.23257 

2.93310 

0.24381 

3-09173 

0.25566 

3.25809 

0.26815 

3.43259 

0.28131 

O.I 

2.75606 

0.46370 

2-90595 

0.48612 

3.06311 

0.50975 

3.22793 

0.53465 

3.40081 

0.56089 

0.15 

2-71331 

0.69198 

2.86088 

0.72542 

3.01560 

0.76069 

3.17787 

0.79785 

3.34807 

0.83701 

O.2 

2.65384 

0.91599 

2.79817 

0.96026 

2.94950 

1.00694 

3.10821 

1.05614 

3.27468 

1.10797 

0.25 

2.57800 

1-13435 

2.71821 

1.18918 

2.86522 

1.24698 

3-01939 

1.30791 

3.18111 

1.37210 

0-3 

2.48627 

1-34571 

2.62149 

1.41076 

2.76327 

1-47934 

2.91196 

1.55162 

3.06792 

1.62777 

°-3S 

2.37922 

1.54878 

2.50862 

1.62365 

2.64429 

1.70258 

2.78657 

1.78576 

2.93582 

1.87341 

0.4 

2.25749 

1.74231 

2.38027 

1.82653 

2.50900 

I-9I53I 

2.64400 

2.00889 

2.78561 

2.10749 

0-45 

2.12185 

1.92509 

2.23725 

2.01814 

2-35824 

2.11624 

2.48513 

2.21964 

2.61823 

2.32858 

o-5 

1.97312 

2.09600 

2.08043 

2.19731 

2.19294 

2.30413 

2.31094 

2.41670 

2.43471 

2.53532 

o-5S 

1.81223 

2.25399 

1.91079 

2.36294 

2.01413 

2.47780 

2.12250 

2.59887 

2.23618 

2.72642 

0.6 

1.64016 

2.39808 

1.72937 

2.51400 

1.82289 

2.63620 

1.92098 

2.76501 

2.02387 

2.90071 

0.65 

1-45799 

2-52739 

1-53728 

2.64956 

1.62042 

2.77835 

1.70761 

2.91409 

1.79907 

3-05713 

0.7 

1.26682 

2.64111 

1-33572 

2.76878 

1.40796 

2.90337 

1.48372 

3.04522 

1.56318 

3.19469 

0-75 

1.06784 

2.73855 

1.12592 

2.87093 

1.  18681 

3.01049 

1.25067 

3.15757 

1.31766 

3.31255 

0.8 

0.86229 

2.81911 

0.90918 

2.95538 

0-95835 

3.09904 

1.00992 

3-25045 

1.06401 

3.41000 

0.85 

0.65141 

2.88229 

0.68684 

3.02161 

0.72398 

3.16850 

0.76294 

3.32330 

0.80380 

3.48641 

o.g 

0.43652 

2.92769 

0.46026 

3.06921 

0.48627 

3.21841 

0.51125 

3.37565 

0.53864 

3-54134 

o-95 

0.21893 

2-955°5 

0.23084 

3.09789 

0.24332 

3.24848 

0.25642 

3.40719 

0.27015 

3-57443 

I.O 

o.oo 

2.96419 

o.oo 

3.10747 

o.oo 

3-25853 

o.oo 

3.41773 

o.oo 

3-58548 

1.05 

0.21893 

2.95505 

0.23084 

3.09789 

0.24332 

3.24848 

0.25642 

3.40719 

0.27015 

3-57443 

i.i 

0.43652 

2.92769 

0.46026 

3.06921 

0.48627 

3.21841 

0.51125 

3.37565 

0.53864 

3-54I34 

i-i5 

0.65141 

2.88229 

0.68684 

3.02161 

0.72398 

3.16850 

0.76294 

3-32330 

0.80380 

3-48641 

1.2 

0.86229 

2.81911 

0.90918 

2.95538 

0.95835 

3.09904 

1.00992 

3-25045 

1.06401 

3.41000 

1-25 

1.06784 

2-73855 

1.12592 

2.87093 

1.  18681 

3.01049 

1.25067 

3.15757 

1.31766 

3-3i255 

i-3 

1.26682 

2.64111 

1-33572 

2.76878 

1.40796 

2.90337 

1.48372 

3.04522 

1.56318 

3.19469 

i-35 

1-45799 

2.52739 

1.53728 

2.64956 

1.62042 

2.77835 

1.70761 

2.91409 

1.79907 

3-05713 

1.4 

1.64016 

2.39808 

1.72937 

2.51400 

1.82289 

2.63620 

1.92098 

2.76501 

2.02387 

2.90071 

1-45 

1.81223 

2-25399 

1.91079 

2.36294 

2.01413 

2.47780 

2.12250 

2.59887 

2.23618 

2.72642 

i-5 

1.97312 

2.09600 

2.08043 

2.19731 

2.19294 

2.30413 

2.31094 

2.41670 

2.43471 

2.53532 

i-SS 

2.12185 

1.92509 

2.23725 

2.01814 

2.35824 

2.11624 

2.48513 

2.21964 

2.61823 

2.32858 

1.6 

2.25749 

1-74231 

2.38027 

1.82653 

2.50900 

I-9I53I 

2.64400 

2.00889 

2.78561 

2.10749 

1.65 

2.37922 

1.54878 

2.50862 

1.62365 

2.64429 

1.70258 

2.78657 

1.78576 

2.93582 

1.87341 

i-7 

2.48627" 

I-3457I 

2.62149 

1.41076 

2.76327 

1-47934 

2.91196 

1.55162 

3.06792 

1.62777 

i-TS 

2.57800 

I-I3435 

2.71821 

1.18918 

2.86522 

1.24698 

3.01939 

1.30791 

3.18111 

1.37210 

1.8 

2.65384 

0.91599 

2.79817 

0.96026 

2.94950 

i  .00694 

3.10821 

1.05614 

3.27468 

1.10797 

1.85 

2-71331 

0.69198 

2.86088 

0.72542 

3.01560 

0.76069 

3.17787 

0.79785 

3.34807 

0.83701 

1.9 

2.75606 

0.46370 

2.90595 

0.48612 

3.06311 

0.50975 

3-22793 

0.53465 

3.40081 

0.56089 

i-95 

2.78181 

0.23257 

2.93310 

0.24381 

3-09173 

0.25566 

3.25809 

0.26815 

3.43259 

0.28131 

2.O 

2.79041 

o.oo 

2.94217 

o.oo 

3.10129 

o.oo 

3.26816 

o.oo 

3.44321 

0.00 

Note. 

Negative  quantities  are  in  heavy  type. 

Examples. 

sinh  (1.85  +  *'  Q-75) 

s=  I.  l868l 

+  »  3.01049. 

sinh  (1.85  +  i  1.35)  =  -1.62042  +  *  2.77835. 


[49] 


TABLE  VII.    HYPERBOLIC  SINES,     sinh  (*  +  ig)  =  u  +  iv.    CONTINUED 


x  =  2.05 


x  =  2.15 


o 
0.05 

O.I 

0.15 

0.2 

3.62686  o.oo 
3.61568  0.29518 
3.58221  0.58854 
3.52666  0.87827 
3.44935  1.16258 

3.81958  o.oo 

3.80781  0.30978 
3.77256  0.61765 
3.71404  0.92172 

3.63264  I.220IO 

4.02186  o.oo 
4.00946  0.32516 

3.97235  0.64831 
3.91074  0.96747 

3.82501  1.28066 

4.23419  o.oo 
4.22113  0.34135 

4.18206  0.68060 
4.11720  1.01564 
4.02695  1.34443 

4.45711  o.oo 
4-44337  0.35839 
4.40223  0.71458 
4-33396  1.06636 
4.23896  1.41156 

0.25 
0-3 

o-3S 
0.4 

0-45 

3.35078  1.43973 
3.23156  1.70800 
3.09241  1.96574 
2.93420  2.21136 
2.75789  2.44335 

3.52883  1.51096 
3.40327  1.79250 
3.23673  2.06299 
3.09OII  2.32076 
2.90443  2.56423 

3.71571  1.58596 
3.58351  1.88148 
3.42920  2.16540 
3.25376  2.43597 
3.05825  2.69152 

3.91188  1.66493 
3.77269  1.97516 

3.61024  2.27322 

3.42553  2.55726 
3.21970  2.82553 

4.11783  1.74806 
3.97131  2.07379 
3.80031  2.38672 
3.60588  2.68495 
3.38921  2.96661 

o-S 
o-SS 
0.6 
0.65 
0.7 

2.56458  2.66027 
2.35546  2.86080 
2.13182  3.04368 
1-89503  3-20780 
1.64656  3-35214 

2.70085  2.79188 
2.48062  3.00232 
2.24509  3.19426 

1-99573  3-36649 
I.73405  3-5I798 

2.84389  2.93048 
2.61199  3-15137 
2.36399  3-35283 
2.10142  3.53361 
1.82589  3.69261 

2.99402  3-07639 
2.74988  3.30828 

2.48879  3-51977 
2.21236  3.70956 
1.92228  3.87647 

3.15165  3.23000 
2.89466  3-47347 
2.61982  3.69552 
2.32883  3-89478 
2.02349  4.07003 

0-75 
0.8 
0.85 
0.9 
0-95 

1-38794  3.4758i 
1.12076  3.57806 
0.84667  3.65825 
0.56737  3-7I587 
0.28456  3-75059 

1.46169  3.64777 
1.18032  3.75507 
0.89166  3.83923 

0.59751  3-8997I 
0.29968  3.93615 

1.53910  3.82885 

1.24282  3.94148 
0.93888  4.02981 
0.62916  4.09329 

0.31555  4-13154 

1.62035  4-01950 
1.30844  4-13773 
0.98845  4.23046 
0.66237  4-29711 
0.33221  4.33726 

1.70566  4.22019 
1-37732  4-34433 
1.04049  4.44170 
0.69724  4.51167 
0.34970  4-55382 

I.O 

1.05 
i.i 
i.iS 

1.2 

o.oo    3.76220 

0.28456  3.75059 
0.56737  3-71587 
0.84667  3.65825 
1.12076  3.57806 

o.oo    3-94832 
0.29968  3.93615 

0-59751  3-8997I 
0.89166  3.83923 
1.18032  3.75507 

o.oo    4.14431 

0.31555  4-13154 

0.62916  4.09329 
0.93888  4.02981 
1.24282  3.94148 

o.oo  4-35067 
0.33221  4.33726 
0.66237  4-29711 
0.98845  4.23046 
1.30844  4.13773 

o.oo    4.56791 

0.34970  4.55382 
0.69724  4.51167 

1.04049  4.44170 
1-37732  4-34433 

I-2S 

i-3 

i  -35 
1.4 

1-45 

1.38794  3-47581 
1.64656  3.35214 
1.89503  3.20780 
2.13182  3.04368 
2.35546  2.86080 

1.46169  3.64777 
1-73405  3-5I798 
1-99573  3-36649 
2.24509  3.19426 
2.48062  3.00232 

1.53910  3.82885 

1.82589  3.69261 
2.10142  3.53361 
2.36399  3-35283 
2.61199  3.15137 

1.62035  4.01950 
1.92228  3.87647 
2.21236  3-70956 
2.48879  3-51977 
2.74988  3.30828 

1.70566  4.22019 
2.02349  4-07003 
2.32883  3-89478 
2.61982  3.69552 
2.89466  3-47347 

i-5 

i-SS 
1.6 
i.6S 
i-7 

2.56458  2.66027 
2.75789  2.44335 

2.93420  2.21136 

3.09241  1.96574 
3.23156  1.70800 

2.70085  2.79188 
2.90443  2.56423 
3.09011  2.32076 
3-23673  2.06299 
3.40327  1.79250 

2.84389  2.93048 
3.05825  2.69152 

3.25376  2.43597 

3.42920  2.16540 

3.58351  1.88148 

2.99402  3.07639 
3.21970  2.82553 
3-42553  2.55726 

3.61024  2.27322 

3.77269  1.97516 

3.15165  3.23000 
3.38921  2.96661 
3.60588  2.68495 
3.80031  2.38672 
3.97131  2.07379 

1-75 
1.8 
1.85 
1.9 

i-9S 

3.35078  1.43973 

3-44935  1.16258 
3.52666  0.87827 
3.58221  0.58854 
3.61568  0.29518 

3.52883  1.51096 

3.63264  I.22OIO 
3.71404  0.92172 
3.77256  0.61765 
3.80781  0.30978 

3.71571  1.58596 
3.82501  1.28066 
3.91074  0.96747 
3.97235  0.64831 
4.00946  0.32516 

3.91188  1.66493 
4.02695  1.34443 

4.11720  1.01564 
4.18206  0.68060 

4.22113  0.34135 

4.11783  1.74806 
4.23896  1.41156 
4.33396  1.06636 
4.40223  0.71458 
4-44337  0.35839 

2.0 

3.62686  o.oo 

3.81958  o.oo 

4.02186  o.oo 

4.23419  o.oo 

4.45711  o.oo 

Note.     Negative  quantities  are  in  heavy  type. 

Examples,  sinh  (2.2  + 

*'  i-o)  =0  +  *  4.56791. 

sinh  (2.2  +  i  1.5)  =  -3.15165  +  i  3.23000. 


[So] 


TABLE  VII.    HYPERBOLIC  SINES,     sinh  (x  +  iq)  =  u  +  iv.     CONTINUED 


q 

X  =  2.2$ 

x  =  2.3 

a:  =  2.35 

x  =  24 

x  =  245 

0 

0.05 

O.I 

0.15 

O.2 

4.69117  o.oo 

4.67671  0.37633 
4.63341  0.75035 
4.56155  I.II974 
4.46157  1.48222 

4.93696  o.oo 

4.92174  0.39522 
4.87618  0.78799 
4.80056  1.17592 
4.69533  1.55659 

5.19510  o.oo 
5.17909  0.41509 
5.13114  0.82761 
5.05156  1.23504 
4.94083  1.63485 

5.46623  o.oo 

5.44938  0.43599 
5.39893  0.86930 
5.31521  1.29724 
5.19869  1.71719 

5.75103  o.oo 

5.73330  0.45799 

5.68022  0.91316 

5.59213  1.36269 
5.46955  1.80383 

0.25 
0.03 

o-3S 
0.4 

o-45 

4.33407  1.83557 
4.17986  2.I776O 
3.99987  2.50620 
3.79523  2.81935 
3.56719  3.II5I2 

4.56116  1.92766 

4.39887  2.28685 
4.20946  2.63194 
3.99409  2.96081 
3.75410  3-27141 

4.79965  2.02457 
4.62887  2.40182 
4.42955  2.76426 
4.20292  3.10966 
3-95038  343588 

5.05014  2.12655 
4.87045  2.52280 
4.66073  2.90349 
4.42228  3.26629 
4.15656  3.60894 

5.31325  2.23385 
5.12420  2.65009 
4-90356  3-04999 

4.65268  3-43I09 
4-373"  3-79I04 

o-5 
0-55 
0.6 
0.65 
0.7 

3.31716  3.39168 
3.04677  3.64734 
2-7574°  3.88050 
2.45II3  4.08975 
2.12975  4.27377 

3.49096  3-56185 
3.20630  3.83034 

2.90188  4.07520 

2.57956  4-29494 

2.24134  4.48820 

3.6735I  3-74093 
3-37395  4-02290 
3.05360  4.28008 
2.71443  4.51087 
2.35853  4-71384 

3.86521  3.92935 
3.55003  4.22554 
3.21297  4-49567 

2.85610  4.73808 

2.48162  4.95127 

4.06659  4.12761 

3-73499  4-43873 
3.38036  4-72249 
3.00490  4.97713 
2.61091  5.20109 

0-75 
0.8 
0.85 
0.9 
0-9S 

1.79523  4.43145 
1.44965  4.56181 
1.09513  4.66404 

0.73386  4-73751 
0.36807  4.78178 

1.88930  4-65378 
1.52560  4-79058 
1.15251  4.89805 
0.77231  4-97521 
0.38735  5-02169 

1.98808  4.88776 

1-60537  5-03I53 
1.21277  5.14429 
0.81269  5.22533 
0.40760  5.27416 

2.09184  5-13394 
1.68916  5.28496 

1.27607  5.40341 
0.85511  5.48853 
0.42888  5-53981 

2.20082  5.39298 
i.777i6  5-55162 
I-34255  5-67603 
0.89966  5.76545 
0.45122  5-8I933 

I.O 

1.05 
i.i 
i.iS 

1.2 

o.oo    4.79657 
0.36807  4.78178 
0.73386  4.73751 
1.09513  4.66404 
1.44965  4.56181 

0.00  5-03722 

0.38735  5-02169 
0.77231  4.97521 
1.15251  4.89805 
1.52560  4-79058 

o.oo    5.29047 
0.40760  5.27416 
0.81269  5.22533 
1.21277  5-14429 
1-60537  5.03153 

o.oo  5-55695 
0.42888  5.53981 
0.85511  5.48853 
1.27607  5.40341 

1.68916  5.28496 

o.oo  5-83732 
0.45122  5.81933 
0.89966  5.76545 
I.34255  5-67603 
1.77716  5.55162 

1.25 

i-3 
i-3S 

1.4 

1-45 

1.79523  443145 
2.12975  4.27377 
2.45113  4.08975 
2-75740  3-88050 
3.04677  3.64734 

1.88930  4-65378 
2.24134  4.48820 

2.57956  4-29494 

2.90188  4.07520 
3.20630  3.83034 

1.98808  4.88776 
2.35853  4-71384 
2.71443  4.51087 
3.05360  4.28008 
3-37395  4-02290 

2.09184  5.13394 
2.48162  4,95127 

2.85610  4.73808 

3.21297  4.49567 
3.55003  4.22554 

2.20082  5.39298 
2.61091  5.20109 
3.00490  4-977I3 
3.38036  4.72249 
3-73499  443873 

i-5 
i-SS 
1.6 
1.65 
i-7 

3.31716  3.39168 
3-56719  3-I][5i2 
3.79523  2.81935 
3.99987  2.50620 
4.17986  2.17760 

3.49096  3.56185 
3.75410  3.27141 

3.99409  2.96081 
4.20946  2.63194 

4.39887  2.28685 

3-6735I  3.74093 
3.95038  3.43588 
4.20292  3.10966 
4-42955  2.76426 
4.62887  2.40182 

3.86521  3.92935 
4.15656  3-60894 

4.42228  3.26629 

4.66073  2.90349 
4.87045  2.52280 

4x16659  4.12761 
4-373II  3.79104 
4.65268  3.43109 
4.90356  3-04999 
5.12420  2.65009 

1-75 
1.8 

1.85 
1.9 

1-95 

4.33407  1.83557 
4.46157  1.48222 

4-56i55  I-II974 
4.63341  0.75035 
4.67671  0.37633 

4.56116  1.92766 

4-69533  1-55659 
4.80056  1.17592 
4.87618  0.78799 
4.92174  0.39522 

4.79965  2.02457 
4.94083  1.63485 
5.05156  1.23504 
5.13114  0.82761 
5.17909  0.41509 

5.05014  2.12655 
5.19869  1.71719 
5.31521  1.29724 
5.39893  0.86930 
5.44938  0.43599 

5.31325  2.23385 
5.46955  1-80383 
5.59213  1.36269 
5.68022  0.91316 
5-73330  0.45799 

2.0 

4.69117  o.oo 

4.93696  o.oo 

5.19510  o.oo 

5.46623  o.oo 

5.75103  o.oo 

Note. 


Negative  quantities  are  in  heavy  type. 


Examples,     sinh  (2.4  +  i  0.05)  =  5.4493s  +  i  0.43599. 

sinh  (2.4  +  i  1.95)  =  -5.44938  +  *  0.43599. 


CsxJ 


TABLE  VII.    HYPERBOLIC   SINES,     sinh  (x  +  iq)  =  u  +  iv.     CONTINUED 


x  =  2.5 


=  2.55 


x  =  2.6 


x  =  2.65 


=  2.7 


o 
0.05 

O.I 

0.15 

O.2 

6.05020  o.oo 
6.03155  0.48113 
5.97572  0.95930 
5.88304  1.43155 
5.75408  1.89498 

6.36451  o.oo 

6.34489  0.50548 

6.28615  1.00784 
6.18866  1.50399 
6.05301  1.99087 

6.69473  o.oo 
6.67409  0.53109 
6.61231  1.05891 

6.50976  1.58019 

6.36706  2.09174 

7.04169  o.oo 
7.01998  0.55803 

6.95500  1.11262 

6.84713  1.66034 
6.69705  2.19784 

7.40626  o.oo 

7-38343  0.58636 
7.31508  1.16911 
7.20163  1.74465 
7.04377  2.30943 

0.25 

o-3 
o-35 
0.4 

o-45 

5.58966  2.34673 
5.39077  2.78401 
5-15865  3.20411 
4.89472  3.60448 
4.60062  3.98260 

5.88004  2.46547 
5.67082  2.92488 
5.42664  3.36624 
5.14900  3.78686 
4.83961  4.18413 

6.18512  2.59039 

5-96505  3-07306 
5.70819  3.53680 
5.41615  3-97872 
5.09071  4-39612 

6.50567  2.72178 
6.27419  3.22894 
6.00403  3.71619 
5.69685  4-18053 

5-35454  4-61910 

6.84249  2.85997 
6.59903  3-39288 
6.31488  3.90488 
5.99179  4-39279 
5.63177  4.85363 

o-S 
o-55 
0.6 
0.65 
0.7 

4.27814  4-33619 
3.92929  4.66304 
3.55622  4-96113 
3.16122  5.22864 
2.74674  5.46392 

4.50039  4.55560 
4.13342  4.89898 
3.74097  5.21217 
3-32545  5-49321 
2.88943  5-74039 

4-73389  4-78641 
4.34788  5-14719 
3-93506  5.47624 

3-49799  5-77I53 
3.03934  6.03123 

4.97923  5.02919 
4.57321  5.40827 
4.13900  5-75401 
3.67927  6.06427 
3.19686  6.33714 

5.23702  5-28454 
4.80998  5.68287 
4-35329  6.04606 
3.86976  6.37218 
3.36237  6.65891 

0-75 
0.8 
0.85 
0.9 
o-9S 

2.31531  5.66550 
1.86961  5.83215 
1.41239  5.96287 
0.94646  6.05680 
0.47469  6.11339 

2-43559  5-952i8 
1.96674  6.12727 
1.48577  6.26458 
0.99563  6.36327 
0.49935  6.42273 

2.56196  6.25374 
2.06878  6.43770 
1.56285  6.58199 
1.04729  6.68567 
0.52526  6.74814 

2.69474  6.57095 
2.17600  6.76424 
1.64385  6.91583 
1.10156  7.02478 
0.55249  7.09042 

2.83425  6.90458 

2.28866  7.10769 
1.72896  7.26697 

1-15859  7.38146 
0.58109  7-45043 

I.O 

1.05 
.1 

•15 

.2 

o.oo    6.13229 
0.47469  6.11339 
0.94646  6.05680 
1.41239  5.96287 
1.86961  5.83215 

o.oo  6.44259 

0.49035  6.42273 
0.99563  6.36327 
1.48577  6.26458 
1.96674  6.12727 

o.oo    6.76901 
0.52526  6.74814 
1.04729  6.68567 
1.56285  6.58199 
2.06878  6.43770 

o.oo  7.11234 
0.55249  7-09042 
1.10156  7.02478 

1.64385  6.91583 

2.17600  6.76424 

o.oo    7-47347 
0.58109  7.45043 
1-15859  7-38146 
1.72896  7.26697 
2.28866  7.10769 

•25 

•3 
•35 
•4 
•45 

2.31531  5-66550 
2.74674  5.46392 
3.16122  5.22864 
3.55622  4.96113 
3.92929  4.66304 

2.43559  5-95218 
2.88943  5.74039 
3.32545  5-49321 
3.74097  5-21217 
4.13342  4.89898 

2.56196  6.25374 
3.03934  6.03123 

3.49799  5-77153 
3.93506  5-47624 
4.34788  5.14719 

2.69474  6.57095 
3.19686  6.33714 
3.67927  6.06427 
4.13900  5.75401 
4.57321  5-40827 

2.83425  6.90458 
3.36237  6.65891 
3.86976  6.37218 
4-35329  6.04606 
4.80998  5.68287 

i-5 
i-SS 
1.6 
1.65 
i-7 

4.27814  4-33619 

4.60062  3.98260 
4.89472  3.60448 
5-15865  3-20411 
5.39077  2.78401 

4-50039  4.55560 
4.83961  4-18413 
5.14900  3.78686 
5.42664  3.36624 
5.67082  2.92488 

'4-73389  4-78641 
5.09071  4-39612 
5.41615  3.97872 
5.70819  3.53680 
5-96505  3-07306 

4.97923  5-02919 

5-35454  4-61910 
5.69685  4-18053 
6.00403  3.71619 
6.27419  3.22894 

5.23702  5.28454 
5.63177  4.85363 
5.99179  4-39279 
6.31488  3.90488 
6.59903  3-39288 

i-75 
1.8 
1.85 
1.9 

1-95 

5.58966  2.34673 

5.75408  1.89498 

5.88304  1.43155 
5.97572  0.95930 
6.03155  0.48113 

5.88004  2.46547 
6.05301  1.99087 
6.18866  J.  50399 
6.28615  1.00784 
6.34489  0.50548 

6.18512  2.59039 

6.36706  2.09174 
6.50976  1.58019 
6.61231  1.05891 
6.67409  0.53109 

6.50567  2.72178 
6.69705  2.19784 
6.84713  1.66034 
6.95500  1.11262 
7.01998  0.55803 

6.84249  2.85997 
7.04377  2.30943 
7.20163  1.74465 
7.31508  1.16911 
7-38343  0.58636 

2.O 

6.05020  o.oo 

6.36451  o.oo 

6.69473  o.oo 

7.04169  o.oo 

7.40626  o.oo 

Note.     Negative  quantities  are  in  heavy  type. 

Examples,  sinh  (2.7  +  *Q-7)  =  3.36237  + 

i  6.65891. 

sinh  (2.5  +  i  1.25)  =  —2.31531  +  i  5.66550. 


[52] 


TABLE  VII.    HYPERBOLIC  SINES,     sinh  (x  +  iq)  =  u  +  iv.    CONTINUED 


x  =  2.75 


x  =  2.85 


x  —  2.9 


x  =  2.95 


o 
0.05 

O.I 

0.15 

0.2 

7-78935  o.oo 
7.76534  0.61616 
7.69345  1.22852 
7.57413  1.83331 
7.40811  2.42680 

8.19192  o.oo 

8.16666  0.64750 
8.09106  1.29101 

7.96557  1.92656 
7.79098  2.55023 

8.61497  o.oo 
8.58841  0.68046 
8.50891  1.35673 
8.37694  2.02463 
8.19332  2.68005 

9.05956  o.oo 

9.03163  0.71512 
8.94802  1.42583 

8.80924  2.12776 
8.61616  2.81656 

9.52681  o.oo 

9.49744  0.75157 
9.40952  1.49851 
9.26358  2.23621 
9.06053  2.96012 

0.25 

o-3 

0.35 
0.4 

0-45 

7.19642  3.00532 
6.94036  3-56531 
6.64151  4-10333 
6.30172  4.61604 
5.92307  5.10030 

7.56834  3.15818 
7.29905  3.74666 
6.98476  4.31204 
6.62740  4.85083 
6.22918  5.35972 

7-959I9  3-31894 
7.67599  3-93738 
7-34547  4-53I53 
6.96966  5.09775 
6.55088  5-63254 

8.36994  3.48800 
8.07213  4-13793 
7.72455  4-76236 
7.32934  5-35742 
6.88894  6.91945 

8.80162  3.66578 
8.48845  4-34884 

8.12294  5.00509 

7-70735  5.63049 
7.24424  6.22116 

o-S 
0-55 
0.6 
0.65 
o-7 

5.50790  5-553" 
5.05878  5.97168 
4.57847  6.35344 
4.06993  6.69602 
3.53629  6.99732 

5.79256  5-83556 
5.32022  6.27542 

4.81509  6.67660 
4.28026  7.03661 

3.71905  7.35323 

6.09170  6.13261 
5.59498  6.59486 
5-06375  7.01646 
4.50131  7-39479 
3.91112  7.72754 

6.40608  6.44498 
5.88371  6.93078 
5.32508  7-37385 
4.73361  7.77146 
4.11295  8.12115 

6.73647  6.77348 

6.18717  7.28404 
5.63048  7.74969 

4-97774  8.16757 
4.32508  8.53508 

0.75 
0.8 
0.85 
0.9 
o-95 

2.98086  7.25548 
2.40704  7.46891 
1.81839  7.63629 
1.21852  7-75659 
0.61115  7.82907 

3.13491  7-62452 
2.53144  7.84881 

1.91237  8.02470 
1.28150  8.15112 
0.64273  8.22728 

3.29681  8.01263 
2.66217  8.24834 

2.OIII2  8.43318 
1.34768  8.56604 
0.67592  8.64608 

3.46694  8.42078 
2.79956  8.66850 
2.11491  8.86275 
1.41723  9.00237 
0.71081  9.08649 

3.64575  8.84998 
2-94395  9.11031 
2.22399  9-3I447 
1.49032  9.46121 
0-74747  9-54962 

I.O 

1.05 
i.i 

1.2 

o.oo    7.85328 
0.61115  7.82907 

1.21852  7.75659 
1.81839  7.63629 
2.40704  7.46891 

o.oo  8.25273 
0.64273  8.22728 
1.28150  8.15112 
1.91237  8.02470 
2.53144  7.84881 

o.oo    8.67281 

O.67592  8.64608 
1.34768  8.56604 
2.0III2  8.43318 
2.66217  8.24834 

o.oo  9.11458 
0.71081  9.08649 

1.41723  9.00237 
2.11491  8.86275 
2.79956  8.66850 

0.00  9-579I5 

0.74747  9-54962 

1.49032  9.46121 

2.22399  9-3I447 

2.94395  9.11031 

1-25 

1-35 
i-4 
i-45 

2.98086  7-25548 
3.53629  6.99732 

4.06993  6.69602 

4.57847  6.35344 
5.05878  5.97168 

3.13491  7-62452 
3-71905  7-35323 

4.28026  7.03661 
4.81509  6.67660 
5.32022  6.27542 

3.29681  8.01263 
3.9III2  7.72754 

4.50131  7-39479 
5-06375  7.01646 
5.59498  6.59486 

3.46694  8.42078 
4.11295  8.12115 
4.73361  7.77146 
5.32508  7-37385 
5.88371  6.93078 

3.64575  8.84998 
4.32508  8.53508 

4-97774  8.16757 
5.63048  7.74969 
6.18717  7.28404 

i-55 
1.6 
1.65 

5-50790  5-553" 
5.92307  5.10030 
6.30172  4.61604 
6.64151  4-10333 
6.94036  3.56531 

5.79256  5-83556 
6.22918  5-35972 

6.62740  4.85083 

6.98476  4.31204 
7.29905  3.74666 

6.09170  6.13261 
6.55088  5.63254 
6.96966  5.09775 
7-34547  4-53I53 
7.67599  3-93738 

6.40608  6.44498 

6.88894  5.91945 
7.32934  5-35742 
7.72455  4.76236 
8.07213  4.13793 

6.73647  6.77348 
7.24424  6.22116 

7-70735  5-63049 
8.12294  5.00509 
8.48845  4.34884 

1-75 
1.8 

1.85 
1.9 
1-95 

7.19642  3.00532 
7.40811  2.42680 
7.57413  1.83331 
7-69345  1.22852 
7-76534  0.61616 

7.56834  3-15818 
7.79098  2.55023 
7-96557  1.92656 

8.09106  1.29101 
8.16666  0.64750 

7.959I9  3-31894 
8.19332  2.68005 
8.37694  2.02463 
8.50891  1.35673 
8.58841  0.68046 

8.36994  3.48800 

8.61616  2.81656 
8.80924  2.12776 
8.94802  1.42583 
9.03163  0.71512 

8.80162  3.66578 
9.06053  2.96012 
9.26358  2.23621 
9.40952  1.49851 
9.49744  0.75157 

2.0 

7.78935  o.oo 

8.19192  o.oo 

8.61497  o.oo 

9.05956  o.oo 

9.52681  o.oo 

Note. 


Negative  quantities  are  in  heavy  type. 


Examples,     sinh  (2.9  +  i  Q-9)  =  1-41723  +  i  9.00237. 

sinh  (2.8  +  i  1.4)  =  —4.81509  +  *  6.67660. 


[53] 


TABLE  VII.    HYPERBOLIC   SINES,     sinh  (*  +  iq)  =  u  +  iv.     CONTINUED 


x  =  3.0 


x  -  3.05 


x  =  3.10 


=  3-iS 


x  =  3.20 


o.o 

10.01787 

o.oo 

10.53399 

o.oo 

11.07645 

o.oo 

11.64661 

o.oo 

12.24588 

o.oo 

0.05 

9.98699 

0.78990 

10.50150 

0.83020 

11.04230 

0.87258 

11.61070 

0.91714 

12.20810 

0.96399 

O.I 

9.89454 

1-57493 

10.40430 

1.65529 

10.94010 

1-73979 

11.50320 

1.82863 

12.09510 

1.92205 

0.15 

9.74108 

2.35025 

10.24290 

2.47017 

10.77040 

2.59626 

11.32480 

2.72885 

11.90750 

2.86826 

0.2 

9-52757 

3.11108 

10.01840 

3.26982 

10.53430 

3.43673 

11.07660 

3.61224 

11.64650 

3.79678 

0.25 

9-2553I 

3-85273 

9.73216 

4.04931 

10.23330 

4.25602 

10.76010 

4-47336 

11.31370 

4.70190 

o-3 

8.92599 

4-57062 

9.38586 

4.80383 

9.86919 

5.04906 

10.37720 

5.30690 

10.91120 

5-57802 

0-35 

8.54164 

5.26034 

8.98171 

5-52874 

9-44423 

5.81097 

9.93036 

6.10773 

10.44130 

6.41976 

0.4 

8.10463 

5.91762 

8.52218 

6.21956 

8.96104 

6-537Q6 

9.42230 

6.87089 

9.90712 

7.22191 

0-45 

7.61765 

6.53842 

8.01011 

6.87204 

8.42260 

7.22284 

8.85615 

7.59170 

9.31184 

7-97954 

o-5 

7.08371 

7.11891 

7.44866 

7.48215 

7.83223 

7.86409 

8.23539 

8.26570 

8.65915 

8.68797 

o.SS 

6.50609 

7-6555I 

6.84128 

8.04612 

7-19358 

8.45686 

7-56387 

8.88873 

7-95306 

9.34284 

0.6 

5.88836 

8.14491 

6.19173 

8.56049 

6.51058 

8.99749 

6.84570 

9-45697 

7-19795 

9.94011 

0.65 

5-23433 

8.58409 

5.50400 

9.02209 

5.78743 

9.48264 

6.08533 

9.96690 

6.39846 

10.47610 

0.7 

4.54806 

8.97035 

4-78233 

9.42805 

5.02860 

9-90933 

5-28745 

10.41540 

5-55952 

10.94750 

0-75 

3-83368 

9.30131 

4.03119 

9-77589 

4.23878 

10.27490 

4.45696 

10.79970 

4.68630 

11.35140 

0.8 

3-09569 

9-57492 

3-255i8 

10.06350 

3.42281 

10.57720 

3-59900 

11.11730 

3.78419 

11.68530 

0.85 

2-33863 

9.78949 

2.45911 

10.28900 

2-58575 

10.81420 

2.71885 

11.36650 

2.85874 

11.94720 

o.g 

1.56714 

9-94371 

1.64788 

10.45110 

1.73274 

10.98460 

1.82193 

11-54550 

1.91568 

12.13540 

o-95 

0.78599 

10.03660 

0.82649 

10.54870 

0.86905 

11.08720 

0.91378 

11.65340 

0.96080 

12.24880 

I.O 

o.oo 

10.06766 

o.oo 

10.58135 

o.oo 

11.12150 

0.00 

11.68946 

o.oo 

12.28665 

1.05 

0.78599 

10.03660 

0.82649 

10.54870 

0.86905 

11.08720 

0.91378 

11.65340 

,  0.96080 

12.24880 

i.i 

1.56714 

9-94371 

1.64788 

10.45110 

1.73274 

10.98460 

1.82193 

11-54550 

1.91568 

12.13540 

•IS 

2.33863 

9.78949 

2.45911 

10.28900 

2.58575 

10.81420 

2.71885 

11.36650 

2.85874 

11.94720 

.2 

3-09569 

9-57492 

3.25518 

10.06350 

3.42281 

10.57720 

3-59900 

11.11730 

3.78419 

11.68530 

•25 

3.83368 

9.30131 

4.03119 

9.77589 

4.23878 

10.27490 

4.45696 

10.79970 

4.68630 

11.35140 

•3 

4.54806 

8.97035 

4.78233 

9.42805 

5.02860 

9-90933 

5.28745 

10.41540 

5.55952 

10.94750 

•35 

5-23433 

8.58409 

5.50400 

9.02209 

5.78743 

9.48264 

6.08533 

9.96690 

6.39846 

10.47610 

•4 

5.88836 

8.14491 

6.19173 

8.56049 

6.51058 

8.99749 

6.84570 

9-45697 

7.19795 

9.94011 

•45 

6.50609 

7-6555I 

6.84128 

8.04612 

7.19358 

8.45686 

7-56387 

8.88873 

7.95306 

9.34284 

•5 

7.08371 

7.11891 

7.44866 

7.48215 

7.83223 

7.86409 

8.23539 

8.26570 

8.65915 

8.68797 

•55 

7.61765 

6.53842 

8.OIOH 

6.87204 

8.42260 

7.22284 

8.85615 

7.59170 

9.31184 

7-97954 

.6 

8.10463 

5.91762 

8.52218 

6.21956 

8.96104 

6.53705 

9.42230 

6.87089 

9.90712 

7.22191 

.65 

8.54164 

5.26034 

8.98171 

5-52874 

9.44423 

5.81097 

9.93036 

6.10773 

10.44130 

6.41976 

•7 

8.92599 

4-57062 

9.38586 

4.80383 

9.86919 

5.04906 

10.37720 

5.30690 

10.91120 

5.57802 

i-75 

9-25531 

3-85273 

9-73216 

4.04931 

10.23330 

4.25602 

10.76010 

4-47336 

11.31370 

4.70190 

1.8 

9.52757 

3.11108 

10.01840 

3.26982 

10.53430 

3-43673 

11.07660 

3.61224 

11.64650 

3.79678 

i.8S 

9.74108 

2.35025 

10.24290 

2.47017 

10.77040 

2.59626 

11.32480 

2.72885 

11.90750 

2.86826 

1.9 

9.89454 

1  -5  7493 

10.40430 

1.65529 

10.94010 

1-73979 

11.50320 

1.82863 

12.09510 

1.92205 

i-95 

9.98699 

0.78990 

10.50150 

0.83020 

11.04230 

0.87258 

11.61070 

0.91714 

12.20810 

0.96399 

2.0 

10.01787 

o.oo 

10.53399 

o.oo 

11.07645 

o.oo 

11.64661 

o.oo 

12.24588 

0.00 

Note. 

Negative  quantities  are  in  heavy  type. 

Examples. 

sinh  (3.0  +  i  0.95)  = 

0.78599  +  i  10.03660. 

sinh  (3.0  +  *  1.05)  =  —  0.78599  +  i  10.03660. 


[54] 


TABLE  VII.     HYPERBOLIC   SINES,     sinh  (x  +  iq)  =  u  +  iv.     CONTINUED 


x  =  3-25 


x  =  3.30 


=  3-35 


x  =  3-4° 


x  =  3-45 


o.o 

12.87578 

O.OO 

13-53788 

o.oo 

14.23382 

o.oo 

14.96536 

o.oo 

I5-73432 

o.oo 

0.05 

12.83610 

1.01326 

13.49615 

1.06507 

14.18950 

1-11953 

14.91923 

1.17679 

15.68581 

1.23699 

O.I 

12.71726 

2.02028 

13.37120 

2.12356 

14.05851 

2.23215 

14.78111 

2.34632 

15.54061 

2.46636 

0.15 

12.52002 

3.01484 

13.16388 

3.16897 

13.84054 

3-33101 

14.55188 

3-50139 

I5-29959 

3.68053 

O.2 

12.24560 

3.99082 

12.87530 

4.19483 

I3-537I7 

4-40933 

14.23291 

4.63487 

14.96423 

4.87198 

0.25 

11.89566 

4.94218 

12.50736 

5-19485 

i3-I5Q33 

5-46075 

13.82620 

5-73977 

14.53662 

6.03341 

o-3 

11.47240 

5.86309 

12.06234 

6.16381 

12.68244 

6-47795 

13-33423 

6.80928 

14.01940 

7-15765 

0-35 

10.97841 

6.74784 

11.54294 

7.09279 

12.13633 

7-45549 

12.76006 

7.83682 

I3-4I572 

8.23775 

0.4 

10.41673 

7-59099 

10.95235 

7-97905 

H-5I54I 

8-38705 

12.10723 

8.81604 

12.72933 

9.26706 

0-45 

9.79082 

8.38733 

10.29428 

8.81610 

10.82349 

9.26691 

n-37975 

9.74090 

11.96447 

10.23924 

o-5 

9-10455 

9.13197 

9-57273 

9.59881 

10.06483 

10.08964 

10.58212 

10.60570 

11.12585 

11.14830 

0-55 

8.36217 

9.82031 

8.79215 

10.32233 

9.24413 

10.85016 

9.71923 

11.40512 

10.21863 

11.98861 

0.6 

7.56820 

10.44810 

7-95737 

10.98222 

8.36643 

n-54379 

8.79642 

12.13423 

9.24840 

12.75501 

0.65 

6.72758 

11.01147 

7-07352 

11.57440 

7-437I5 

12.16623 

7-81938 

12.78852 

8.22116 

13.44278 

0.7 

S-84548 

11.50695 

6.14670 

12.09520 

6.46202 

12.71369 

6.79414 

I3-36397 

7.14324 

14.04764 

0-75 

4-92735 

11.93150 

5.18072 

12.54145 

5-44705 

13.18275 

5.72700 

13.85702 

6.02127 

14.56595 

0.8 

3-97883 

12.28247 

4-18343 

12.91035 

4-39850 

I3-57054 

4-62455 

14.26465 

4.86217 

14.99442 

0.85 

3-00579 

12.55773 

3.16036 

13.19970 

3.32282 

13.87465 

3-49359 

14.58432 

3.67311 

15-33045 

0.9 

2.01422 

12.75556 

2.11780 

13.40764 

'2.22666 

14.09323 

2.34110 

14.81410 

2.46139 

15-57196 

o-95 

I.OIO22 

12.87474 

1.06217 

13-5329° 

1.11677 

14.22498 

1.17417 

14.95250 

1.23450 

15.71746 

I.O 

o.oo 

12.91456 

o.oo 

I3-57476 

0.00 

14.26891 

o.oo 

14.99874 

0.00 

15.76607 

1.05 

I.OIO22 

12.87474 

1.06217 

13-53290 

1.11677 

14.22498 

1.17417 

14.95250 

1.23450 

15.71746 

i.i 

2.01422 

12.75556 

2.11780 

13.40764 

2.22666 

14.09323 

2.34110 

14.81410 

2.46139 

15-57196 

i-iS 

3-00579 

12.55773 

3.16036 

13.19970 

3-32282 

13.87465 

3-49359 

14.58432 

3.67311 

15-33045 

1.2 

3-97883 

12.28247 

4.18343 

12.91035 

4-39850 

I3-57054 

4-62455 

14.26465 

4.86217 

14.99442 

1-25 

4-92735 

11.93150 

5.18072 

12.54145 

5.44705 

13.18275 

5.72700 

13.85702 

6.02127 

14.56595 

!-3 

5.84548 

11.50695 

6.14670 

12.09520 

6.46202 

12.71369 

6.79414 

I3-36397 

7.14324 

14.04764 

i-3S 

6.72758 

11.01147 

7.07352 

11.57440 

7-43715 

12.16623 

7.81938 

12.78852 

8.22116 

13.44278 

1.4 

7.56820 

10.44810 

7-95737 

10.98222 

8.36643 

n-54379 

8.79642 

12.13423 

9.24840 

12.75501 

i  -45 

8.36217 

9.82031 

8.79215 

10.32233 

9.24413 

10.85016 

9.71923 

11.40512 

10.21863 

11.98861 

i-S 

9-10455 

9.13197 

9-57273 

9.59881 

10.06483 

10.08964 

10.58212 

10.60570 

11.12585 

11.14830 

i-SS 

9.79082 

8.38733 

10.29428 

8.81610 

10.82349 

9.26691 

n-37975 

9.74090 

11.96447 

10.23924 

1.6 

10.41673 

7-59099 

10.95235 

7-97905 

11-51541 

8.38705 

12.10723 

8.81604 

12.72933 

9.26706 

1.65 

10.9784! 

6.74784 

11.54294 

7.09279 

12.13633 

7-45549 

12.76006 

7.83682 

13.41572 

8.23775 

i-7 

11.47240 

5.86309 

12.06234 

6.16381 

12.68244 

6-47795 

I3-33423 

6.80928 

14.01940 

7-15765 

i-75 

11.89566 

4.94218 

12.50736 

5-I9485 

13-15033 

5-46075 

13.82620 

5-73977 

14.53662 

6.03341 

1.8 

I2.2456O 

3.99082 

12.87530 

4.19483 

13-53717 

4.40933 

14.23291 

4-63487 

14.96423 

4.87198 

1.85 

I2.52O02 

3.01484 

13.16388 

3.16897 

13.84054 

3-33101 

14.55188 

3-50I39 

15.29959 

3-68053 

1.9 

12.71726 

2.02028 

13.37120 

2.12356 

14.05851 

2.23215 

14.78111 

2.34632 

15.54061 

2.46636 

i-95 

12.83610 

1.01326 

13.49615 

1.06507 

14.18950 

I.II953 

14.91923 

1.17679 

15.68581 

1.23699 

2.O    12.87578   O.OO 


13.53788  o.oo 


14.23382  o.oo    14.96536  o.oo    15.73432  o.oo 


Note. 

Examples,     sinh  (3.40  +  *  o)  =  14.96536  +  i  o. 

sinh  (3.45  +  i  1.45)  =  —10.21863  +  i  11.98861. 


-/ 


TABLE  VII.    HYPERBOLIC   SINES,     sinh  (x  +  iq)  =  u  +  iv.     CONTINUED 


x  =  3-50 


=  3-55 


x  =  3.60 


=  3-65 


x  =  3.70 


0.0 

16.54263 

o.oo 

17.39230 

o.oo 

18.28546 

o.oo 

19.22434 

0.00 

20.21129 

o.oo 

0.05 

16.49163 

1.30029 

17-33870 

1.36684 

18.22900 

1.43680 

19.16506 

1.51036 

20.14900 

1.58770 

O.I 

16.33896 

2.59256 

17.17817 

2.72525 

18.06033 

2.86475 

18.98765 

3.01142 

19.96246 

3.16561 

0.15 

16.08555 

3-86885 

16.91175 

4.06686 

17.78022 

4-27503 

18.69316 

4.49390 

19.65290 

4.72401 

0.2 

15-733°° 

5.12129 

16.54105 

5-38339 

17.39050 

5.65896 

18.28342 

5.94868 

19.22208 

6.25327 

0.2S 

15.28340 

6.34215 

16.06840 

6.66674 

16.89356 

7.00800 

17.76096 

7.36678 

18.67280 

7-74399 

o-3 

14.73960 

7-5239I 

15.49665 

7.90898 

16.29246 

8.31383 

17.12900 

8.73946 

18.00840 

9.18696 

t>-35 

14.10490 

8.65928 

14.82933 

9.10246 

15.59090 

9.56840 

16.39143 

10.05827 

17.23295 

10.57330 

0.4 

13-3833° 

9.74126 

14.07066 

10.23982 

14.79326 

10.76400 

15-55281 

11.31505 

16.35127 

11.89444 

o-4S 

12.57910 

10.76320 

13.22520 

11.31405 

13.90436 

11.89320 

14.61830 

12.50208 

15.36878 

13.14223 

o-5 

11.69740 

11.71820 

12.29820 

12.31852 

12.92978 

12.94910 

13-59365 

13.61203 

14.29155 

14.30902 

0-55 

JO-74357 

12.60210 

11.29540 

13.24705 

11-87545 

13-92515 

12.48520 

14.63806 

13.12620 

15.38760 

0.6 

9.72361 

13.40770 

10.22293 

14.09390 

10.74789 

14.81535 

11.29972 

!5-57383 

11.87990 

16.37127 

0.65 

8.64360 

14.13065 

9.08745 

14.85386 

9-55412 

15.61421 

10.04469 

16.41360 

10.56037 

17.25404 

0.7 

7.51020 

14.76650 

7-89594 

15.52225 

8.30143 

16.31680 

8.72766 

17.15216 

9-17573 

18.03040 

0-75 

6.33°59 

i5-3"3o 

6.65574 

16.09492 

6.99754 

16.91880 

7.35683 

17.78497 

7-73453 

18.69565 

0.8 

S-n^S 

15.76170 

5-37451 

16.56840 

5-65052 

17.41650 

5.94065 

18.30814 

6.24563 

19.24560 

0.85 

3.86180 

16.11491 

4.06015 

16.93970 

4.26865 

17.80680 

4.48783 

18.71843 

4.71823 

19.67690 

0.9 

2.58783 

16.36878 

2.72075 

17.20653 

2.86047 

18.08732 

3-00735 

19.01331 

3.16174 

19.98688 

o.QS 

1.29792 

16.52173 

1.36458 

17.36731 

1.43466 

18.25632 

1.50832 

19.19100 

I-58576 

20.17362 

I.O 

0.00 

16.57282 

0.00 

17.42102 

o.oo 

18.31278 

o.oo 

19-25033 

o.oo 

20.23601 

1.05 

1.29792 

16.52173 

1.36458 

17.36731 

1.43466 

18.25632 

1.50832 

19.19100 

1.58576 

20.17362 

i.i 

2.58783 

16.36878 

2.72075 

17.20653 

2.86047 

18.08732 

3-00735 

19.01331 

3.16174 

19.98688 

i-iS 

3.86180 

16.11491 

4.06015 

16.93970 

4.26865 

17.80680 

4.48783 

18.71843 

4.71823 

19.67690 

1.2 

5.11195 

15.76170 

5.37451 

16.56840 

5.65052 

17.41650 

5.94065 

18.30814 

6.24563 

19.24560 

1-25 

6.33059 

15-3113° 

6.65574 

16.09492 

6.99754 

16.91880 

7.35683 

17.78497 

7-73453 

18.69565 

i-3 

7.51020 

14.76650 

7.89594 

15.52225 

8.30143 

16.31680 

8.72766 

17.15216 

9-17573 

18.03040 

i-3S 

8.64360 

14.13065 

9.08745 

14.85386 

9-55412 

15.61421 

10.04469 

16.41360 

10.56037 

17.25404 

1.4 

9.72361 

13.40770 

10.22293 

14.09390 

10.74789 

14.81535 

11.29972 

I5-57383 

11.87990 

16.37127 

i.  45 

10.74357 

12.60210 

11.29540 

13.24705 

11.87545 

13.92515 

12.48520 

14.63806 

13.12620 

15.38760 

i-5 

11.69740 

11.71820 

12.29820 

12.31852 

12.92978 

12.94910 

13.59365 

13.61203 

14-29155 

14.30902 

i-55 

12.57910 

10.76320 

13.22520 

11.31405 

13-90436 

11.89320 

14.61830 

12.50208 

15.36878 

13.14223 

1.6 

13-38330 

9.74126 

14.07066 

10.23982 

14.79326 

10.76400 

15.55281 

11-31505 

16.35127 

11.89444 

i.6S 

14.10490 

8.65928 

14.82933 

9.10246 

15-59090 

9.56840 

16.39143 

•10.05827 

17.23295  ^10.57330 

i-7 

14.73960 

7-52391 

15-49665 

7.90898 

16.29246 

8.31383 

17.12900 

8.73946 

18.00840 

9.18696 

i-75 

15.28340 

6-34215 

16.06840 

6.66674 

16.89356 

7.00800 

17.76096 

7.36678 

18.67280 

7-74399 

1.8 

15.73300 

5.12129 

16.54105 

5-38339 

17.39050 

5.65896 

18.28342 

5.94868 

19.22208 

6.25327 

1.85 

16.08555 

3-86885 

16.91175 

4.06686 

17.78022 

4.27503 

18.69316 

4.49390 

19.65290 

4.72401 

1.9 

16.33896 

2.59256 

17.17817 

2.72525 

18.06033 

2.86475 

18.98765 

3.01142 

19.96246 

3.16561 

i-95 

16.49163 

1.30029 

17.33870 

1.36684 

18.22900 

1.43680 

19.16506 

1.51036 

20.14900 

1.58770 

2.O 

16.54263 

0.00 

17.39230 

o.oo 

18.28546 

0.00 

19.22434 

o.oo 

20.21129 

0.00 

Note.     Negative  quantities  are  in  heavy  type. 

Examples,  sinh  (3.70  +  i  0.5)  =  14.29155  -4- 

i  14.30902 

. 

- 

sinh  (3.70  +  i  1.5)  =  —14.29155  +  *  14.30902. 


[56] 


TABLE  VII.    HYPERBOLIC   SINES,     sinh  (x  +  iq)  =  u  +  iv.     CONTINUED 


=  3-75 


x  =  3.80 


*  =  3.85 


x  =  3.90 


=  3-95 


o.o 

21.24878 

o.oo 

22.33941 

o.oo 

23.48589 

o.oo 

24.69110 

o.oo 

25.95806 

o.oo 

0.05 

21.18327 

1.66900 

22.27052 

1.75448 

23.41348 

1.84435 

24.61500 

1.93883 

25.87805 

2.03816 

O.I 

20.98716 

3.32772 

22.06437 

3-49815 

23-19673 

3.67733 

24.38710 

3-86571 

25.63849 

4-06375 

0.15 

20.66167 

4.96592 

21.72216 

5.22025 

22.83696 

5.48764 

24.00888 

5-76875 

25.24084 

6.06429 

O.2 

20.20879 

6.57350 

21.24603 

6.91017 

22.33640 

7.26411 

23.48262 

7.63623 

24.68760 

8.02744 

0.25 

19.63131 

8.14055 

20.63891 

8.55748 

21.69813 

8.99580 

22.81160 

9.45662 

23.98212 

9.94109 

o-3 

18.93280 

9.65741 

19-90455 

10.15203 

20.92608 

10.67203 

21-99993 

11.21871 

23.12881 

11.79366 

°-3S 

18.11756 

11.11473 

19.04746 

11.68400 

20.02501 

12.28246 

21.05272 

12.91164 

22.13290 

13-57315 

0.4 

17.19062 

12.50353 

18.07296 

13-14392 

19.00048 

13.81716 

I9-97556 

14.52497 

21.00052 

15.26910 

o.4S 

16.15770 

13.81524 

16.98701 

14.52281 

17.85880 

15.26668 

18.77526 

16.04874 

19.73867 

16.87098 

o-5 

15.02516 

15.04177 

15-79634 

15.81216 

16.60702 

16.62208 

17.45924 

17-47355 

18.35512 

18.36873 

0-55 

13.79998 

16.17556 

14.50828 

17.00402 

15.25286 

17.87500 

16.03558 

18.79065 

16.85842 

19.75331 

0.6 

12.48971 

17.20963 

13.13076 

18.09105 

13.80465 

19.01770 

14-51307 

19.99190 

15.25776 

21.01610 

0.65 

11.10246 

18.13760 

11.67230 

19.06655 

12.27134 

20.04315 

12.90106 

21.06988 

13.56305 

22.14931 

0.7 

9.64674 

18.95373 

10.14188 

19.92448 

10.66233 

20.94503 

11.20952 

22.01797 

11.78474 

23.14597 

0.75 

8.13156 

19.65301 

8.54892 

20.65958 

8.98766 

21.71778 

9.44887 

22.83030 

9-93373 

23.99991 

0.8 

6.56624 

20.23113 

6.90325 

21.26731 

7-25754 

22.35664 

7.62997 

23.50188 

8.02149 

24.70590 

0.85 

4.96043 

20.68452 

5-21503 

21.74391 

5.48267 

22.85766 

.5.76402 

24.02856 

6.05979 

25.25957 

0.9 

3-32404 

21.01038 

3-49465 

22.08646 

3.67400 

23.21775 

3.86254 

24.40710 

4.06074 

25.65749 

o-9S 

1.66716 

21.20670 

1-75273 

22.29283 

1.84268 

23.43470 

1.93724 

24.63516 

2.03665 

25.89724 

I.O 

o.oo 

21.27230 

o.oo 

22.36178 

o.oo 

23.50717 

o.oo 

24-71135 

o.oo 

25-97731 

1.05 

1.66716 

21.20670 

1.75273 

22.29283 

1.84268 

23.43470 

1.93724 

24.63516 

2.03665 

25.89724 

i.i 

3-32404 

21.01038 

3.49465 

22.08646 

3.67400 

23.21775 

3.86254 

24.40710 

4.06074 

25-65749 

i-iS 

4.96043 

20.68452 

5-21503 

21.74391 

5.48267 

22.85766 

5.76402 

24.02856 

6.05979 

25.25957 

1.2 

6.56624 

20.23113 

6.90325 

21.26731 

7.25754 

22.35664 

7.62997 

23.50188 

8.02149 

24.70590 

1-25 

8.13156 

19.65301 

8.54892 

20.65958 

8.98766 

21.71778 

9.44887 

22.83030 

9-93373 

23.99991 

i-3 

9.64674 

18.95373 

10.14188 

19.92448 

10.66233 

20.94503 

11.20952 

22.01797 

11.78474 

23.14597 

1-35 

11.10246 

18.13760 

11.67230 

19.06655 

12.27134 

20.04315 

12.90106 

21.06988 

13-56305 

22.14931 

1.4 

12.48971 

17.20963 

13.13076 

18.09105 

13.80465 

19.01770 

14.51307 

19.99190 

15-25776 

21.01610 

1-45 

13.79998 

16.17556 

14.50828 

17.00402 

15.25286 

17.87500 

16.03558 

18.79065 

16.85842 

19.75331 

i-5 

15.02516 

15.04177 

15-79634 

15.81216 

16.60702 

16.62208 

17.45924 

17-47355 

18.35512 

18.36873 

1-55 

16.15770 

13.81524 

16.98701 

14.52281 

17.85880 

15.26668 

18.77526 

16.04874 

19.73867 

16.87098 

1.6 

17.19062 

12.50353 

18.07296 

13-14392 

19.00048 

13.81716 

19.97556 

14.52497 

21.00052 

15.26910 

1-65 

18.11756 

11.11473 

19.04746 

11.68400 

20.02501 

12.28246 

21.05272 

12.91164 

22.13290 

13.57315 

i-7 

18.93280 

9.65741 

19.90455 

10.15203 

20.92608 

10.67203 

21.99993 

11.21871 

23.12881 

11.79366 

1-75. 

19.63131 

8.14055 

20.63891 

8.55748 

21.69813 

8.99580 

22.81160 

9.45662 

23.98212 

9.94109 

1.8 

20.20879 

6.57350 

21.24603 

6.91017 

22.33640 

7.26411 

23.48262 

7.63623 

24.68760 

8.02744 

1.85 

20.66167 

4.96592 

21.72216 

5.22025 

22.83696 

5.48764 

24.00888 

5.76875 

25.24084 

6.06429 

1.9 

20.98716 

3.32772 

22.06437 

3.49815 

23-19673 

3.67733 

24.38710 

3-86571 

25.63849 

4.06375 

i  -95 

21.18327 

1.66900 

22.27052 

1.75448 

23.41348 

1.84435 

24.61500 

1.93883 

25.87805 

2.03816 

2.0 

21.24878 

o.oo 

22.33941 

o.oo 

23.48589 

0.00 

24.69110 

o.oo 

25.95806 

o.oo 

Note.     Negative  quantities  are  in  heavy  type. 

Examples,  sinh  (3.90  +  fr'o-75)  = 

9.44887  + 

i  22.83030. 

sinh  (3.95  +  i  1.95)  =  -25.87805  +  i  2.03816. 


[57] 


TABLE  VIII.     HYPERBOLIC   COSINES,     cosh  (x  +  iq)  =  u  +  iv 


x  =  0.05 


x  =  0.15 


0 

I.OOOO   O.OO 

1.00125  o.oo 

1.00500  o.oo 

1.01127  o.oo 

1.02007  o.oo 

0.05 

0.99692  o.oo 

0.99816  0.00392 

1.00191  0.00786 

1.00815  0.01181 

1.01692  0.01580 

O.I 

0.98769  o.oo 

0.98892  0.00783 

0.99263  0.01567 

0.99882  0.02355 

1.00751  0.03150 

0.15 

0.97237  o.oo 

0.97385  0.01168 

0.97724  0.02338 

0.98333  0.03515 

0.99188  0.04700 

O.2 

0.95106  o.oo 

0.95225  0.01546 

0.95582  0.03095 

0.96178  0.04653 

0.97014  0.06222 

0.2S 

0.92388  o.oo 

0.92503  0.01914 

0.92850  0.03833 

0.93429  0.05762 

0.94242  0.07705 

0-3 

0.89101  o.oo 

0.89208  0.02271 

0.89547  0.04547 

0.90105  0.06835 

0.90889  0.09141 

o-3S 

0.85264  o.oo 

0.85367  0.02614 

0.85691  0.05234 

0.86225  0.07867 

0.86975  0.10520 

0.4 

0.80902  o.oo 

0.81000  0.02940 

0.81307  0.05888 

0.81814  0.08850 

0.82525  0.11834 

0-45 

0.76041  o.oo 

0.76133  0.03249 

0.76421  0.06505 

0.76898  0.09778 

0.77567  0.13076 

o.S 

0.70711  o.oo 

0.70796  0.03537 

0.71065  0.07083 

0.71508  0.10646 

0.72130  0.14237 

o-SS 

0.64945  o.oo 

0.65023  0.03804 

0.65270  0.07617 

0.65677  0.11449 

0.66248  0.15310 

0.6 

0.58779  o.oo 

0.58850  0.04047 

0.59027  0.08104 

0.59441  0.12181 

0.59958  0.16288 

0.65 

0.52250  o.oo 

0.52313  0.04265 

0.52511  0.08541 

0.52839  0.12838 

0.53298  0.17167 

0.7 

0.45399  o.oo 

0.45439  0.04457 

0.45626  0.08925 

0.45911  0.13415 

0.46310  0.17939 

o.7S 

0.38268  o.oo 

0.38316  0.04621 

0.38460  0.09254 

0.38700  0.13910 

0.39036  o.i  8601 

0.8 

0.30902  o.oo 

0.30940  0.04757 

0.31056  0.09526 

0.31250  0.14319 

0.31522  0.19148 

0.85 

0.23345  o.oo 

0.23374  0.04864 

0.23461  0.09740 

0.23608  0.14640 

0.23813  0.19577 

0.9 

0.15643  o.oo 

0.15663  0.04941 

0.15722  0.09893 

0.15820  0.14871 

0.15957  0.19886 

o-9S 

0.07846  o.oo 

0.07856  0.04987 

0.07885  0.09986 

0.07934  0.15010 

0.08003  0.20072 

I.O 

o.oo    o.oo 

o.oo    0.05002 

o.oo    0.10017 

o.oo    0.15056 

o.oo    0.20134 

1-05 

0.07846  o.oo 

0.07856  0.04987 

0.07885  0.09986 

0.07934  0.15010 

0.08003  0.20072 

i.i 

0.15643  o.oo 

0.15663  0.04941 

0.15722  0.09893 

0.15820  0.14871 

0.15957  0.19886 

i-iS 

0.23345  o.oo 

0.23374  0.04864 

0.23461  0.09740 

0.23608  0.14640 

0.23813  0.19577 

1.2 

0.30902  o.oo 

0.30940  0.04757 

0.31056  0.09526 

0.31250  0.14319 

0.31522  0.19148 

1-25 

0.38268  o.oo 

0.38316  0.04621 

0.38460  0.09254 

0.38700  0.13910 

0.39036  0.18601 

i-3 

0.45399  o.oo 

0.45439  0.04457 

0.45626  0.08925 

0.45911  0.13415 

0.46310  0.17939 

•35 

0.52250  o.oo 

0.52313  0.04265 

0.52511  0.08541 

0.52839  0.12838 

0.53298  0.17167 

•4 

0.58779  o.oo 

0.58850  0.04047 

0.59027  0.08104 

0.59441  0.12181 

0.59958  0.16288 

•45 

0.64945  o.oo 

0.65023  0.03804 

0.65270  0.07617 

0.65677  0.11449 

0.66248  0.15310 

•5 

0.70711  o.oo 

0.70796  0.03537 

0.71065  0.07083 

0.71508  0.10646 

0.72130  0.14237 

•55 

0.76041  o.oo 

0.76133  0.03249 

0.76421  0.06505 

0.76898  0.09778 

0.77567  0.13076 

.6 

0.80902  o.oo 

0.81000  0.02940 

0.81307  0.05888 

0.81814  0.08850 

0.82525  0.11834 

•65 

0.85264  o.oo 

0.85367  0.02614 

0.85691  0.05234 

0.86225  0.07867 

0.86975  0.10520 

1-7 

0.89101  o.oo 

0.89208  0.02271 

0.89547  O-O4547 

0.90105  0.06835 

0.90889  0.09141 

1-75 

0.92388  o.oo 

0.92503  0.01914 

0.92850  0.03833 

0.93429  0.05762 

0.94242  0.07705 

1.8 

0.95106  o.oo 

0.95225  0.01546 

0.95582  0.03095 

0.96178  0.04653 

0.97014  0.06222 

1.85 

0.97237  o.oo 

0.97385  0.01168 

0.97724  0.02338 

0.98333  0.03515 

0.99188  0.04700 

1.9 

0.98769  o.oo 

0.98892  0.00783 

0.99263  0.01567 

0.99882  0.02355 

1.00751  0.03150 

1-95 

0.99692  o.oo 

0.99816  0.00392 

1.00191  0.00786 

1.00815  0.01181 

1.01692  0.01580 

I.OOOO        O.OO          I.OOI25     O.OO 


1.00500   o.oo 


1.01127  o.oo 


1.02007  o.oo 


Note. 


Negative  quantities  are  in  heavy  type. 


Examples,    cosh  (o  +  i  0.75)  =  0.38268  +  «  o. 

cosh  (0.2  +  i  1.5)  =  —0.72130  +  *  0.14237 


[58] 


TABLE  VIII.     HYPERBOLIC   COSINES,     cosh  (x  +  iq)  =  u  +  iv.     CONTINUED 


x  =  0.25 


x  =  0.3 


x  =  0.35 


x  =  0.4 


x  =  0.45 


0 

o.os 

O.I 

0.15 

0.2 

1.03141  o.oo 
1.02823  0.01982 
1.01871  0.03952 
1.00292  0.05897 
0.98093  0.07806 

1.04534  o.oo 
1.04212  0.02389 
1.03247  0.04764 
1.01646  0.07109 
0.99418  0.09410 

i.  06  1  88  o.oo 
1.05860  0.02803 
1.04880  0.05588 
1.03254  0.08338 
1.00991  0.11038 

1.08107  o.oo 

1.07774  0.03223 

1.06776  0.06426 
1.05120  0.09589 
1.02816  0.12693 

1.10297  o.oo 

1.09957  0.03651 

1.08939  0.07280 
1.07250  0.10863 
1.04899  0.14380 

0.25 
0-3 
0-35 
0.4 

0-4S 

0.95290  0.09667 
0.91900  0.11468 
0.87942  0.13199 
0.83443  0.14848 
0.78429  0.16406 

0.96577  0.11654 
0.93140  0.13825 
0.89130  0.15911 
0.84570  0.17899 
0.79488  0.19777 

0.98105  0.13669 
0.94614  0.16216 
0.90540  0.18663 
0.85908  0.20995 
0.80746  0.23198 

0.99878  0.15719 

0.96324  0.18648 
0.92177  0.21462 
0.87461  0.24143 
0.82206  0.26676 

1.01901  0.17808 
0.98275  0.21126 

0.94044  0.24314 
0.89232  0.27352 
0.83871  0.30222 

0-5 
0-55 

0.6 
0.65 
0.7 

0.72932  0.17862 
0.66985  0.19208 
0.60625  °.  2043  7 
0.53891  0.21539 
0.46825  0.22508 

0.73917  0.21533 
0.67889  0.23156 
0.61444  0.24636 
0.54619  0.25965 
0.47457  0.27133 

0.75086  0.25257 
0.68964  0.27161 
0.62416  0.28897 
0.55483  0.30455 
0.48208  0.31826 

0.76443  0.29045 
0.70210  0.31234 
0.63544  0.33231 

0.56486  0.35022 
0.49080  0.36598 

0.77992  0.32905 
0.71632  0.35385 
0.64831  0.37647 
0.57630  0.39677 

0.50074  0.41462 

o.7S 
0.8 
0.85 
0.9 
o-9S 

0.39471  0.23338 
0.31872  0.24025 
0.24078  0.24563 
0.16135  0.24950 
0.08092  0.25183 

0.40003  0.28134 
0.32303  0.28962 
0.24403  0.29611 
0.16353  0.30077 
0.08202  0.30358 

0.40636  0.33000 
0.32814  0.33971 
0.24789  0.34732 
0.16611  0.35279 
0.08331  0.35609 

0.41371  0.37949 
0.33407  0.39065 
0.25237  0.39940 

0.16912  0.40570 
0.08482  0.40949 

0.42209  0.42992 

0.34084  0.44257 
0.25748  0.45249 
0.17254  0.45961 

0.08654  0.46391 

I.O 

1.05 
i.i 

1.2 

o.oo    0.25261 
0.08092  0.25183 
0.16135  0.24950 
0.24078  0.24563 
0.31872  0.24025 

o.oo    0.30452 
0.08202  0.30358 

0.16353  0.30077 

0.24403  0.29611 
0.32303  0.28962 

o.oo    0.35719 
0.08331  0.35609 
o.  16611  0.35279 
0.24789  0.34732 
0.32814  0.33971 

o.oo  0.41075 
0.08482  0.40949 
0.16912  0.40570 

0.25237  0.39940 
0.33407  0.39065 

o.oo  0.46534 
0.08654  0.46391 

0.17254  0.45961 
0.25748  0.45249 
0.34084  0.44257 

1-25 

i-35 

1.4 

1-45 

0.39471  0.23338 

0.46825  0.22508 

0.53891  0.21539 

0.60625  0.20437 
0.66985  0.19208 

0.40003  0.28134 

0.47457  0.27133 
0.54619  0.25965 

0.61444  0.24636 
0.67889  0.23156 

0.40636  0.33000 
0.48208  0.31826 
0.55483  0.30455 
0.62416  0.28897 
0.68964  0.27161 

0.41371  0.37949 

0.49080  0.36598 
0.56486  0.35022 

0.63544  0.33231 

0.70210  0.31234 

0.42209  0.42992 
0.50074  0.41462 

0.57630  0.39677 
0.64831  0.37647 
0.71632  0.35385 

i-SS 
1.6 

1.65 

0.72932  0.17862 
0.78429  0.16406 
0.83443  0.14848 
0.87942  0.13199 
0.91900  0.11468 

0.73917  0.21533 
0.79488  0.19777 
0.84570  0.17899 
0.89130  0.15911 

0.93140  0.13825 

0.75086  0.25257 
0.80746  0.23198 
0.85908  0.20995 
0.90540  0.18663 
0.94614  0.16216 

0.76443  0.20945 

0.82206  0.26676 
0.87461  0.24143 
0.92177  0.21462 
0.96324  0.18648 

0.77992  0.32905 
0.83871  0.30222 

0.89232  0.27352 
0.94044  0.24314 
0.98275  0.21126 

i-75 
1.8 
1.85 
1.9 
I.9S 

0.95290  0.09667 
0.98093  0.07806 
1.00292  0.05897 
1.01871  0.03952 
1.02823  0.01982 

0.96577  0.11654 

0.99418  0.09410 
1.01646  0.07109 
1.03247  0.04764 
1.04212  0.02389 

0.98105  0.13669 
1.00991  0.11038 
1.03254  0.08338 
1.04880  0.05588 
1.05860  0.02803 

0.99878  0.15719 

1.02816  0.12693 
1.05120  0.09589 
1.06776  0.06426 

1.07774  0.03223 

1.01901  0.17808 
1.04899  0.14380 
1.07250  0.10863 
1.08939  0.07280 

1.09957  0.03651 

2.0 

1.03141  o.oo 

1.04534  o.oo 

i.  06  1  88  o.oo 

1.08107  o.oo 

1.10297  o/oo 

Note. 

Negative  quantities  are  in  heavy  type. 

Examples,  cosh  (0.3  +  i  0.9)  =  0.16353  + 

i  0.30077. 

cosh  (0.45  +  i  1.7)  =  —0.98275  +  i  0.21126. 


[59] 


TABLE  VTII.     HYPERBOLIC   COSINES,     cosh  (x  +  iq)  =  u  +  iv.     CONTINUED 


0.5 


x  =  0.55 


x  =  0.6 


x  =  0.65 


x  —  0.7 


o 
0.05 

O.I 

0.15 

0.2 

1.12763  o.oo 
1.12415  0.04088 
1.11374  0.08152 
1.09647  0.12165 
1.07244  0.16103 

1.15510  o.oo 

1.15154  0.04536 
1.14088  0.09044 
1.12319  0.13497 
1.09857  0.17866 

1.18547  o.oo 
1.  18181  0.04995 
1.17087  0.09959 
1.15271  0.14862 
1.12744  0.19674 

1.21879  o.oo 
1.21504  0.05467 
1.20379  0.10900 
1.18512  0.16265 

1.15912  0.21531 

1.25517  o.oo 
1.25130  0.05952 
1.23972  0.11867 
1.22049  0.17709 

1.19374  0.23442 

0.25 
0-3 
0-35 
0.4 

0.45 

1.04179  0.19942 
1.00472  0.23657 
0.96146  0.27227 
0.91227  0.30629 

0.85745  0.33842 

1.06717  0.22125 
1.02920  0.26248 
0.98489  0.30208 

0.93450  0.33983 
0.87835  0.37548 

1.09523  0.24364 
1.05626  0.28904 
1.01078  0.33265 
0.95906  0.37422 
0.90144  0.41347 

1.12602  0.26663 

1.08595  0.31632 
1.03919  0.36405 

0.98602  0.40954 

0.92678  0.45250 

1.15962  0.29030 

1.11836  0.34439 
1.07021  0.39636 

1.01545  0.44589 

0.95444  0.49266 

0-5 
0-55 

0.6 
0.65 
0.7 

0.79735  0.36847 
0.73233  0.39624 

0.66280  0.42158 

0.58918  0.44431 
0.51193  0.46430 

0.81678  0.40882 
0.75018  0.43963 
0.67895  0.46773 

0.60354  0.49296 

0.52441  0.51514 

0.83825  0.45018 
0.76990  0.48412 
0.69680  0.51506 
0.61940  0.54284 

0.53819  0.56726 

0.86182  0.49268 

0.79154  0.52981 
0.71639  0.56368 
0.63682  0.59408 
0.55332  0.62081 

0.88754  0.53640 
0.81517  0.57683 
0.73777  0.61371 

0.65582  0.64680 
0.56984  0.67590 

0-75 
0.8 
0.85 
0.9 
0-9.5 

0.43152  0.48143 
0.34846  0.49559 

0.26324  0.50670 
0.17640  0.51468 
0.08847  0.51949 

0.44204  0.53414 
0.35695  0.54986 
0.26965  0.56218 
0.18070  0.57103 
0.09063  0.57637 

0.45366  0.58819 
0.36633  0.60549 
0.27674  0.61906 
0.18545  0.62882 
0.09301  0.63469 

0.46641  0.64371 
0.37663  0.66265 

0.28452  0.67750 

0.19066  0.68817 
0.09563  0.69460 

0.48033  0.70084 
0.38787  0.72146 
0.29301  0.73763 
0.19635  0.74925 

0.09848  0.75625 

1.0 

1.05 
i.i 

1.2 

o.oo    0.52110 
0.08847  0.51949 
0.17640  0.51468 
0.26324  0.50670 

0.34846  0.49559 

o.oo  0.57815 
0.09063  0.57637 
0.18070  0.57103 
0.26965  0.56218 

0.35695  0.54986 

o.oo    0.63665 
0.09301  0.63469 
0.18545  0.62882 
0.27674  0.61906 
0.36633  0.60549 

o.oo    0.69675 
0.09563  0.69460 
0.19066  0.68817 
0.28452  0.67750 
0.37663  0.66265 

o.oo    0.75858 
0.09848  0.75625 

0.19635  0.74925 
0.29301  0.73763 

0.38787  0.72146 

1.25 
1.3 

1-35 
1.4 

i-45 

0.43152  0.48143 
0.51193  0.46430 
0.58918  0.44431 
0.66280  0.42158 

0.73233  0.39624 

0.44204  0.53414 
0.52441  0.51514 

0.60354  0.49296 

0.67895  0.46773 
0.75018  0.43963 

0.45366  0.58819 
0.53819  0.56726 
0.61940  0.54284 
0.69680  0.51506 
0.76990  0.48412 

0.46641  0.64371 
0.55332  0.62081 
0.63682  0.59408 

0.71639  0.56368 

0.79154  0.52981 

0.48033  0.70084 
0.56984  0.67590 
0.65582  0.64680 
0.73777  0.61371 
0.81517  0.57683 

i-SS 
1.6 
1.65 
1-7 

0.79735  0.36847 
0.85745  0.33842 

0.91227  0.30629 
0.96146  0.27227 
1.00472  0.23657 

0.81678  0.40882 
0.87835  0.37548 
0.93450  0.33983 
0.98489  0.30208 
1.02920  0.26248 

0.83825  0.45618 
0.90144  0.41347 
0.95906  0.37422 

1.01078  0.33265 

1.05626  0.28904 

0.86182  0.49268 
0.92678  0.45250 
0.98602  0.40954 
1.03919  0.36405 
1.08595  0.31632 

0.88754  0.53640 
0.95444  0.49266 
1.01545  0.44589 
1.07021  0.39636 

1.11836  0.34439 

i!s 

1.85 
1.9 
1-95 

1.04179  0.19942 
1.07244  0.16103 
1.09647  0.12165 
1.11374  0.08152 
1.12415  0.04088 

1.06717  0.22125 
1.09857  0.17866 
1.12319  0.13497 
1.14088  0.09044 

1.15154  0.04536 

1.09523  0.24364 

1.12744  0.19674 
1.15271  0.14862 

1.17087  0.09959 

1.  18181  0.04995 

1.12602  0.26663 

1.15912  0.21531 

1.18512  0.16265 
1.20379  0.10900 
1.21504  0.05467 

1.15962  0.29030 
1.19374  0.23442 
1.22049  0.17709 
1.23972  0.11867 
1.25130  0.05952 

2.0 

•  1.12763  o.oo 

1.15510  o.oo 

1.18547  o.oo 

1.21879  o.oo 

1.25517  o.oo 

Note.  Negative  quantities  are  in  heavy  type. 

Examples,     cosh  (0.6  +  i  0.95)  =  0.09301  +  i  0.63469. 

cosh  (0.6  +  i  1.05)  =  —0.09301  +  i  0.63469. 


[60] 


TABLE  VIII.     HYPERBOLIC  COSINES,     cosh  (x  +  iq)  =  u  +  iv.     CONTINUED 


0.75 


0.8 


x  =  0.85 


x  =  o.g 


x  =  0.95 


o 

0.05 

O.I 

0.15 

O.2 

1.29468  o.oo 
1.29069  0.06452 
1.27874  0.12864 
1.25891  0.19197 
1.23132  0.25411 

1.33743  o.oo 
I-3333I  0.06968 
1.32097  0.13893 
1.30048  0.20732 
1.27198  0.27444 

1.38353  o.oo 
1.37927  0.07502 
1.36650  0.14957 
1.34530  0.22320 
1.31582  0.29546 

1.43309  o.oo 
1.42867  0.08054 
1.41544  0.16058 

1.39349  0.23964 
1.36294  0.31721 

1.48623  o.oo 
1.48164  0.08627 
1.46793  0.17200 
1.44516  0/25667 
1.41348  0.33976 

0.25 

°-3 
0-35 
0.4 

0-45 

1.19613  0.31469 
1.15356  0.37332 
1.10390  0.42966 
1.04742  0.48335 
0.98449  0.53405 

1.23563  0.33986 
1.19166  0.40319 
1.14035  0.46403 
1.08201  0.52202 
1.01700  0.57678 

1.27822  0.36589 
1.23274  0.43407 
1.17965  0.49957 
1.11930  0.56199 
1.05205  0.62095 

1.32400  0.39283 

1.27689  0.46603 

1.22194  0.53635 
1.15939  0.60337 

1.08973  0.66667 

1.37309  0.42076 
1.32425  0.49916 
1.26722  0.57448 

1.20238  0.64626 
1.13013  0.71406 

o-5 
0-55 
0.6 
0.65 
0.7 

0.91548  0.58147 
0.84083  0.62529 
0.76100  0.66527 
0.67647  0.70114 
0.58777  0.73269 

0.94571  0.62799 
0.86859  0.67532 
0.78613  0.71849 
0.69881  0.75724 
0.60718  0.79131 

0.97830  0.67608 
0.89853  0.72704 
0.81322  0.77351 
0.72290  0.81522 
0.62811  0.85191 

1.01334  0.72586 
0.93071  0.78057 
0.84235  0.83047 
0.74879  0.87525 

0.65061  0.91463 

1.05092  0.77745 
0.96523  0.83605 
0.87358  0.88950 
0.77655  0.93747 
0.67473  0.97965 

0.75 
0.8 
0.85 
0.9 
o.95 

0.49545  0.75972 
0.40008  0.78207 
0.30224  0.79960 
0.20253  0.81219 
0.10158  0.81978 

0.51182  0.82050 
0.41329  0.84464 
0.31222  0.86357 
0.20922  0.87717 
0.10493  0.88537 

0.52945  0.88334 
0.42753  0.90932 
0.32298  0.92970 
0.21643  0.94435 
0.10855  0.95317 

0.54842  0.94838 
0.44285  0.97628 
0-33455  0.99816 
0.22418  1.01388 
0.11244  1-02335 

0.56875  1.01579 
0.45927  1.04567 

0.34695  1.06911 
0.23250  1.08595 
0.11661  1.09610 

.0 

•05 
.1 

•IS 

.2 

o.oo    0.82232 
0.10158  0.81978 
0.20253  0.81219 
0.30224  0.79960 
0.40008  0.78207 

o.oo    0.88811 
0.10493  0.88537 
0.20922  0.87717 
0.31222  0.86357 
0.41329  0.84464 

o.oo    0.95612 

0.10855  0.95317 
0.21643  0.94435 
0.32298  0.92970 
0.42753  0.90932 

o.oo    1.02652 
0.11244  1-02335 
0.22418  1.01388 
0.33455  0.99816 
0.44285  0.97628 

o.oo    1.09948 
o.  11661  1.09610 
0.23250  1.08595 
0.34695  1.06911 
0.45927  1.04567 

•25 

•3 
i-35 
1.4 

1-45 

0.49545  0.75972 
0.58777  0.73269 
0.67647  0.70114 
0.76100  0.66527 
0.84083  0.62529 

0.51182  0.82050 
0.60718  0.79131 
0.69881  0.75724 
0.78613  0.71849 
0.86859  0.67532 

0.52945  0.88334 

0.62811  0.85191 
0.72290  0.81522 
0.81322  0.77351 
0.89853  0.72704 

0.54842  0.94838 
0.65061  0.91463 
0.74879  0.87525 
0.84235  0.83047 
0.93071  0.78057 

0.56875  1.01579 
0.67473  0.97965 
0.77655  0.93747 
0.87358  0.88950 
0.96523  0.83605 

i-5 
1-55 
1.6 

1.65 
i-7 

0.91548  0.58147 
0.98449  0.53405 
1.04742  0.48335 
1.10390  0.42966 
I.I5356  0.37332 

0.94571  0.62799 
1.01700  0.57678 
1.08201  0.52202 
1.14035  0.46403 
1.19166  0.40319 

0.97830  0.67608 
1.05205  0.62095 
1.11930  0.56199 

1.17965  0.49957 
1.23274  0.43407 

1.01334  0.72586 
1.08973  0.66667 
I-I5939  0.60337 
1.22194  0.53635 
1.27689  0.46603 

1.05092  0.77745 
1.13013  0.71406 
1.20238  0.64626 
1.26722  0.57448 
1.32425  0.49916 

1-75 
1.8 
1.85 
1.9 

1-95 

1.19613  0.31469 
1.23132  0.25411 
1.25891  0.19197 
1.27874  0.12864 
1.29069  0.06452 

1.23563  0.33986 
1.27198  0.27444 
1.30048  0.20732 
1.32097  0.13893 
1.33331  0.06968 

1.27822  0.36589 

1.31582  0.29546 
1.34530  0.22320 
1.36650  0.14957 
1.37927  0.07502 

1.32400  0.39283 
1.36294  0.31721 
1-39349  0.23964 
1.41544  0.16058 
1.42867  0.08054 

1.37309  0.42076 
1.41348  0.33976 
1.44516  0.25667 
1.46793  0.17200 
1.48164  0.08627 

2.0 

1.29468  o.oo 

1.33743  0.00 

1.38353  o.oo 

1.43309  o.oo 

1.48623  o.oo 

Note. 

Negative  quantities  are  in  heavy  type. 

Examples,  cosh  (0.9  + 

i  i.o)  =  o  +  *  1.02652. 

cosh  (0.9  +  i  i.io)  =  —0.22418  +  i  1.01388. 


[61] 


TABLE  VIII.     HYPERBOLIC  COSINES,     cosh  (x  +  iq)  =  u  +  iv.     CONTINUED 


x  =  1.05 


0 

0.05 

O.I 

0.15 

O.2 

1.54308  o.oo 

1.53832  0.09221 
1.52408  0.18384 

1.50045  0.27435 
1.46756  0.36316 

1.60379  o.oo 
1.59885  0.09838 
1.58405  0.19615 
1.55948  0.29271 
1.52530  0.38746 

1.66852  o.oo 
1.66337  0.10479 
1.64798  0.20894 
1.62242  0.31180 
1.58685  0.41274 

1.73741  o.oo 
1.73206  0.11147 
1.71602  0.22226 
1.68941  0.33167 
1.65238  0.43904 

i.  81066  o.oo 
1.80507  0.11843 
1.78836  0.23613 
1.76063  0.35238 

1.72204  0.46645 

0.25 
0.3 

o-3S 
0.4 

o-4S 

1.42562  0.44973 
1.37490  0.53353 
1.31569  0.61404 
1.24838  0.69077 
1.17337  0.76323 

1.48171  0.47983 
1.42899  0.56924 
1.36746  0.65514 
1.29750  0.73700 
1.21953  0.81432 

1.54151  0.51113 
1.48666  0.60637 
1.42265  0.69787 
1.34986  0.78508 
1.26875  0.86743 

1.60517  0.54371 
1.54805  0.64502 
1.48139  0.74235 
1.40560  0.83511 
1.32114  0.92272 

1.67283  0.57765 
1.61331  0.68528 
1.54384  0.78869 
1.46485  0.88724 
1.37684  0.98032 

o-S 
o.SS 
0.6 
0.65 
0.7 

1.09112  0.83099 

1.00215  0.89363 

0.90700  0.95076 

0.80626  I.OO2O2 

0.70055  1-04711 

1.13405  0.88661 
1.04158  0.95344 
0.94269  1.01439 
0.83798  1.06909 
0.72811  1.11719 

1.17982  0.94445 
1.08362  1.01564 
0.98073  1.08056 
0.87180  1.13883 
0.75749  1.19007 

1.22854  1.00464 
1.12836  1.08037 
1.02123  I-I4943 
0.90780  1.21141 
0.78877  1.26592 

1.28033  1-06735 
1.17593  1.14781 

1.06428  I.  22118 
0.94607  1.28703 
0.82202  1.34494 

0-75 
0.8 
0.85 
0.9 
o-9S 

0.59051  1.08574 
0.47684  1.11768 
0.36023  1.14273 
0.24139  1.16073 
0.12107  1.17158 

0.61375  1.15841 
0.49560  1.19249 
0.37440  1.21921 
0.25089  1.23842 
0.12583  1.24999 

0.63852  1.23398 
0.51560  1.27028 
0.38951  1.29875 
0.26101  1.31920 
0.13091  1.33153 

0.66488  1.31263 
0.53689  1.35124 
0.40559  1.38152 
0.27179  1.40329 
0.13632  1.41640 

0.69291  1.39456 
0.55952  1.43558 

0.42269  1.46776 
0.28325  1.49088 
0.14206  1.50481 

I.O 

1.05 
i.i 
i-iS 

1.2 

o.oo    1.17520 
0.12107  1.17158 
0.24139  1.16073 
0.36023  1.14273 
0.47684  1.11768 

o.oo    1.25386 
0.12583  1.24999 
0.25089  1.23842 
0.37440  1.21921 
0.49560  1.19249 

o.oo    1.33565 
0.13091  1.33153 

0.26101  1.31920 

0.38951  1.29875 
0.51560  1.27028 

o.oo    1.42078 
0.13632  1.41640 
0.27179  1.40329 
0.40559  1.38152 
0.53689  1.35124 

o.oo    1.50946 
0.14206  1.50481 
0.28325  1.49088 
0.42269  1.46776 

0.55952  1.43558 

1.25 

i-3 
1-35 
1.4 

i.4S 

0.59051  1.08574 
0.70055  1.04711 

O.8O626   1.00202 
O.9O7OO  0.95076 
I.002I5  0.89363 

0.61375  1.15841 

0.72811  1.11719 
0.83798  1.06909 
0.94269  1.01439 

1.04158  0.95344 

0.63852  1.23398 
0.75749  1.19007 
0.87180  1.13883 
0.98073  1.08056 
1.08362  1.01564 

0.66488  1.31263 
0.78877  1.26592 
0.90780  1.21141 

I.02I23  I.I4943 
1.12836  1.08037 

0.69291  1.39456 

0.82202  1.34494 
0.94607  1.28703 
1.06428  I.  22118 

1.17593  1.14781 

i.S 
i-SS 
1.6 
1.65 
1.7 

.09112  0.83099 
.17337  0.76323 
.24838  0.69077 
.31569  0.61404 
.37490  0.53353 

1.13405  0.88661 
1.21953  0.81432 
1.29750  0.73700 
1.36746  0.65514 
1.42899  0.56924 

1.17982  0.94445 
1.26875  0.86743 
1.34986  0.78508 
1.42265  0.69787 
1.48666  0.60637 

1.22854  1.00464 
I.32II4  0.92272 
1.40560  0.835II 
I.48I39  0.74235 
1.54805  0.64502 

1.28033  1.06735 

1.37684  0.98032 
1.46485  0.88724 
1.54384  0.78869 
1.61331  0.68528 

1-75 
1.8 
1.85 
1.9 
1.95 

1.42562  0.44973 
X.46756  0.36316 
1.50045  0.27435 
1.52408  0.18384 
1.53832  O.O922I 

1.48171  0.47983 
1.52530  0.38746 
1.55948  0.29271 
1.58405  0.19615 
1.59885  0.09838 

1.54151  0.51113 
1.58685  0.41274 
1.62242  0.31180 
1.64798  0.20894 
1.66337  0.10479 

I.605I7  0.54371 
1.65238  0.43904 
I.6894I  0.33167 
I.7I6O2  O.22226 
I.732O6  O.III47 

1.67283  0.57765 

1.72204  0.46645 
1.76063  0.35238 
1.78836  0.23613 
1.80507  0.11843 

2.O 

1.54308  o.oo 

1.60379  o.oo 

1.66852  o.oo 

I«7374i  o.oo 

i.  81066  o.oo 

Note. 

Negative  qua 

n  tities  are  in  heavy 

type. 

Examples,    cosh  (1.2  4-  i  o)  =  i. 81066  +  i  o. 

cosh  (E.I  +  »  i.i)  =  —0*26101  +  i  1.31920. 


[62] 


TABLE  VIII.     HYPERBOLIC   COSINES,     cosh  (*  +  iq)  =  u  +  iv.     CONTINUED 


9 

x  =  1.25 

*=  1.3 

x  =  i-35 

x  =  1.4 

x  =  1.45 

o 
0.05 

O.I 

o-iS 

O.2 

1.88842  o.oo 
1.88260  0.12569 
1.86517  0.25060 

.83624  0.37396 

.79600  0.49502 

1.97091  o.oo 

1.96484  1.13325 

1.94665  0.26569 
1.91646  0.39648 

1.87445  0.52483 

2.05833  o.oo 
2.05200  0.14116 
2.03299  0.28144 
2.00146  0.41999 
1-95759  0.55595 

2.15090  o.oo 
2.14427  0.14941 
2.12442  0.29790 

2.09147  0.44455 
2.04562  0.58846 

2.24884  o.oo 
2.24191  0.15804 
2.22115  0.31510 
2.18671  0.47022 
2.13878  0.62244 

0.25 
0-3 
0-35 
0.4 

0-45 

.74467  0.61303 

.68260  0.72726 
.61014  0.83700 

1.52777  0.94158 
1.43597  1.04036 

1.82089  0.64994 

1.75610  0.77105 
1.68048  0.88740 
1.59451  0.99829 

1.49870  1.10301 

1.90165  0.68848 
1.83399  0.81677 
1.75502  0.94002 
1.66523  1.05748 
1.56517  1.16842 

1.98717  0.72875 

1.91646  0.86454 
1.83394  0.99500 

1.74012  1.11932 

1.63556  1.23674 

2.07766  0.77083 
2.00373  0.91446 
1.91745  1.05245 
1.81935  1.18396 
1.71007  1.30817 

o-5 
o.SS 
0.6 
0.65 
0.7 

1.33532  1.13273 

1.22643  1.2x811 
1.10999  1.29598 
0.98670  1.36586 

0.85733  1-42732 

1.39365  1.20094 

1.28001  1.29146 

1.15848  1.37402 

1.02980  1.44811 

0.89478  1.51327 

1.45546  1.27215 
1.33678  1.36804 
1.20986  1.45550 
1.07548  1.53398 
0.93446  1.60300 

1.52092  1.34655 
1.39690  1.44804 
1.26427  1.54061 
1.12384  1.62369 
0.97649  1.69675 

1.59017  1.42431 
1.46051  1.53166 
1.32184  1.62958 
1.17502  1.71745 

1.02095  1-79473 

0.75 
0.8 
0.85 
0.9 
o-95 

0.72267  1.47998 

0.58355  1-52352 
0.44084  1.55766 
0.29541  1.58220 

0.14816  1.59698 

0.75424  1.56910 
0.60905  1.61526 
0.46010  1.65146 

0.30832  1.67747 
0.15464  1.69315 

0.78769  1.66215 
0.63606  1.71104 
0.48051  1.74938 
0.32199  1.77694 
0.16150  1.79354 

0.82311  1.75934 

0.66466  1.81110 
0.50212  1.85169 
0.33647  1.88086 
0.16876  1.89843 

0.86060  1.86094 
0.69493  1.91569 
0.52498  1.95862 
0.35180  1.98947 
0.17644  2.00806 

I.O 

1.05 
i.i 
i-iS 

1.2 

o.oo    1.60192 
0.14816  1.59698 
0.29541  1.58220 

0.44084  1.55766 
0.58355  1-52352 

o.oo  1.69838 
0.15464  1.69315 
0.30832  1.67747 

0.46010  1.65146 
0.60905  1.61526 

o.oo    1.79909 
0.16150  1.79354 
0.32199  1.77694 
0.48051  1.74938 
0.63606  1.71104 

o.oo    1.90430 
0.16876  1.89843 
0.33647  1.88086 

0.50212  1.85169 
0.66466  1.81110 

o.oo    2.01427 
0.17644  2.00806 
0.35180  1.98947 
0.52498  1.95862 
0.69493  1.91569 

I.2S 

i-3 
i-35 
1.4 

i-4S 

0.72267  1.47998 
0.85733  1.42732 

0.98670  1.36586 

.10999  1-29598 

.22643  1.  21811 

0.75424  1.56910 
0.89478  1.51327 

1.02980  1.44811 

1.15848  1.37402 

1.28001  1.29146 

0.78769  1.66215 
0.93446  1.60300 

1.07548  1.53398 

1.20986  1.45550 

1.33678  1.36804 

0.82311  1.75934 
0.97649  1.69675 
1.12384  1.62369 
1.26427  1.54061 
1.39690  1.44804 

0.86060  1.86094 

1.02095  1.79473 
1.17502  1.71745 

1.32184  1.62958 
1.46051  1.53166 

i-5 
•55 
.6 

.65 
•7 

.33532  1.13273 

•43597  1.04036 
•52777  0.94158 
.61014  0.83700 
.68260  0.72726 

1-39365  1.20094 

1.49870  1.10301 
1.59451  0.99829 
1.68048  0.88740 
1.75610  0.77105 

1.45546  1.27215 

1.56517  1.16842 
1.66523  1.05748 
1.75502  0.94002 
1.83399  0.81677 

1.52092  1.34655 
1.63556  1.23674 

1.74012  1.11932 
1.83394  0.99500 
1.91646  0.86454 

1.59017  1.42431 

1.71007  1.30817 

1.81935  1.18396 
1.91745  1.05245 

2.00373  0.91446 

•75 
.8 

•85 
1.9 

1-95 

.74467  0.61303 
.79600  0.49502 
.83624  0.37396 
.86517  0.25060 
.88260  0.12569 

1.82089  0.64994 

1.87445  0.52483 

1.91646  0.39648 
1.94665  0.26569 

1.06484  0.13325 

1.90165  0.68848 
X-95759  0.55595 
2.00146  0.41999 
2.03299  0.28144 
2.05200  0.14116 

1.98717  0.72875 
2.04562  0.58846 
2.09147  0.44455 

2.12442  0.29790 

2.14427  0.14941 

2.07766  0.77083 
2.13878  0.62244 
2.18671  0.47022 
2.22115  0.31510 
2.24191  0.15804 

2.O  1.88842     O.OO 


1.97091    o.oo 


2.05833   o.oo 


2.15090   o.oo 


2.24884  o.oo 


Note.  Negative  quantities  are  in  heavy  type. 

Examples,    cosh  (1.4  +  i  1.9)  =  —2.12442  +  *  0.29790. 
cosh  (1.4  +  i  1.4)  =  —1.26427  +  *  1.54061. 


[63] 


TABLE  VIII.     HYPERBOLIC  COSINES,     cosh  (x  +  iq)  =  u  +  iv.     CONTINUED 


*=  1.5 


=  i-55 


x  =  1.6 


x  =  1.65 


x  =  1.7 


o 

2.35241  o.oo 

2.46186  o.oo 

2.57746  o.oo 

2.69952  o.oo 

2.82832  o.oo 

0.05 

2.34516  0.16706 

2.45427  0.17650 

2.56952  0.18639 

2.69121  0.19673 

2.81960  0.20757 

O.I 

2.32345  0.33309 

2.43155  0.35192 

2-54573  0.37162 

2.66629  °-39225 

2.79350  0.41387 

0.15 

2.28742  0.49707 

2.39384  0.52516 

2.50625  0.55456 

2.62494  0.58536 

2.75017  0.61761 

O.2 

2.23727  0.65798 

2.34137  0.69517 

2.45131  0.73409 

2.56740  0.77485 

2.68989  0.81754 

0.25 

2.17334  0.81484 

2.27446  0.86089 

2.38127  0.90909 

2.49404  0.95957 

2.61302  1.01244 

°-3 

2.09601  0.96667 

2.19353  1.02130 

2.29654  1.07848 

2.40529  1.13837 

2.52005  1.20109 

0.35 

2.00576  1.11255 

2.09908  1.17542 

2.19765  1.24123 

2.30173  1.31015 

2.41154  1.38234 

0.4 

1.90314  1.25156 

1.99169  1.32229 

2.08522  1.39632 

2.18396  1.47385 

2.28816  1.55506 

o-4S 

1.78879  1.38286 

1.87201  1.46101 

1.95992  1.54281 

2.05274  1.62847 

2.15067  1.71820 

o-S 

1.66341  1.50563 

1.74080  1.59071 

1.82254  1.67978 

1.00885  1.77305 

1.99992  1.87074 

0-55 

1.52777  1.61912 

1.59885  1.71062 

1.67393  1.80640 

1.75320  1.90669 

1.83684  2.01175 

0.6 

1.38271  1.72263 

1.44704  1.81997 

1.51500  1.92187 

1.58674  2.02858 

1.66244  2.14036 

0.65 

1.22913  1.81551 

1.28632  1.91811 

1.34672  2.02550 

1.41050  2.13797 

1.47779  2.25577 

0.7 

1.06797  1.89720 

1.11766  2.00442 

1.17015  2.11664 

1.22556  2.23417 

1.28403  2.35727 

0-75 

0.90023  1.96720 

0.94211  2.07837 

0.98636  2.19473 

1.03306  2.31660 

1.08235  2.44424 

0.8 

0.72693  2.02507 

0.76076  2.13951 

0.79648  2.25930 

0.83420  2.38474 

0.87400  2.51614 

0.85 

0.54916  2.07045 

0.57471  2.18745 

0.60170  2.30993 

0.63019  2.43818 

0.66026  2.57253 

0.9 

0.36800  2.10307 

0.38512  2.22191 

0.40320  2.34632 

0.42230  2.47659 

0.44245  2.61306 

0-95 

0.18457  2.12272 

0.19316  2.24268 

0.20223  2.36824 

0.21180  2.49973 

0.22191  2.63747 

I.O 

o.oo    2.12928 

o.oo    2.24961 

o.oo    2.37557 

0,00    2.50747 

o.oo    2.64563 

i.  05 

0.18457  2.12272 

0.19316  2.24268 

0.20223  2.36824 

o.  21180  2.49973 

0.22191  2.63747 

i.i 

0.36800  2.10307 

0.38512  2.22191 

0.40320  2.34632 

0.42230  2.47659 

0.44245  2.61306 

i-iS 

0.54916  2.07045 

0.57471  2.18745 

0.60170  2.30993 

0.63019  2.43818 

0.66026  2.57253 

1.2 

0.72693  2.02507 

0.76076  2.13951 

0.79648  2.25930 

0.83420  2.38474 

0.87400  2.51614 

I.2S 

0.90023  1.96720 

0.94211  2.07837 

0.98636  2.19473 

1.03306  2.31660 

1.08235  2.44424 

i-3 

1.06797  1.89720 

1.11766  2.00442 

1.17015  2.11664 

1.22556  2.23417 

1.28403  2.35727 

i-35 

1.22913  1.81551 

1.28632  1.91811 

1.34672  2.02550 

1.41050  2.13797 

1.47779  2.25577 

1.4 

1.38271  1.72263 

1.44704  1.81997 

1.51500  1.92187 

1.58674  2.02858 

1.66244  2.14036 

i-4S 

1-52777  1.61912 

1.59885  1.71062 

1-67393  1.80640 

1-75320  1.90669 

1.83684  2.01175 

i-5 

1.66341  1.50563 

1.74080  1.59071 

1.82254  1.67978 

1.90885  1.77305 

1.99992  1.87074 

i-SS 

1.78879  1.38286 

1.87201  1.46101 

1.95992  1.54281 

2.05274  1.62847 

2.15067  1.71820 

1.6 

1.90314  1.25156 

1.99169  1.32229 

2.08522  1.39632 

2.18396  1.47385 

2.28816  1.55506 

1.65 

2.00576  1.11255 

2.09908  1.17542 

2.19765  1.24123 

2.30173  1.31015 

2.41154  1.38234 

i-7 

2.09601  0.96667 

2.19353  1.02130 

2.29654  1.07848 

2.40529  1.13837 

2.52005  1.20109 

1-75 

2-17334  0.81484 

2.27446  0.86089 

2.38127  0.90909 

2.49404  0.95957 

2.61302  1.01244 

1.8 

2.23727  0.65798 

2.34137  0.69517 

2.45131  0.73409 

2.56740  0.77485 

2.68989  0.81754 

1.85 

2.28742  0.49707 

2.39384  0.52516 

2.50625  0.55456 

2.62494  0.58536 

2.75017  0.61761 

1.9 

2.32345  0-33309 

2.43155  0.35192 

2-54573  0.37162 

2.66629  0.39225 

2.79350  0.41387 

i.  95 

2.34516  0.16706 

2.45427  0.17650 

2.56952  0.18639 

2.69121  0.19673 

2.81960  0.20757 

2.35241   o.oo 


2.46186   o.oo 


2.57746   o.oo 


2.69952   o.oo 


2.82832   o.oo 


Note. 


Negative  quantities  are  in  heavy  type. 


Examples,     cosh  (1.6  +  i  0.4)  =  2.08522  -f-  i  1.39632. 

cosh  (1.7  +  *  1.2)  =  —0.87400  +  i  2.51614. 


[64] 


TABLE  VIII.     HYPERBOLIC   COSINES,     cosh  (x  +  iq)  =  u  +  n>.     CONTINUED 


? 

x  = 

i-75 

x  = 

1.8 

X  = 

1.85 

X  = 

1.9 

x  = 

i-95 

o 

2.96419 

o.oo 

3.10747 

o.oo 

3-25853 

o.oo 

3-41773 

o.oo 

3-58548 

o.oo 

0.05 

2.95505 

0.21893 

3.09789 

0.23084 

3.24848 

0.24332 

3.40719 

0.25642 

3-57443 

0.27015 

O.I 

2.92769 

0.43652 

3.06921 

0.46026 

3.21841 

0.48627 

3-37565 

0.51125 

3-54I34 

0.53864 

0.15 

2.88229 

0.65141 

3.02161 

0.68684 

3.16850 

0.72398 

3-32330 

0.76294 

3.48641 

0.80380 

O.2 

2.81911 

0.86229 

2.95538 

0.90918 

3.09904 

0.95835 

3-25045 

1.00992 

3.41000 

1.06401 

0.25 

2.73855 

1.06784 

2.87093 

1.12592 

3.01049 

1.  18681 

3-I57S7 

1.25067 

3-31255 

1.31766 

o-3 

2.64111 

1.26682 

2.76878 

1-33572 

2.90337 

1.40796 

3.04522 

1.48372 

3.19469 

1.56318 

0-35 

2-52/39 

1.45799 

2.64956 

1.53728 

2.77835 

1.62042 

2.91409 

1.90761 

3-057I3 

1.79907 

0.4 

2.39808 

1.64016 

2.51400 

1.72937 

2.63620 

1.82289 

2.76501 

1.92098 

2.90071 

2.02387 

0-45 

2-25399 

1.81223 

2.36294 

1.91079 

2.47780 

2.01413 

2.59887 

2.12250 

2.72642 

2.23618 

o-5 

2.09600 

1.97312 

2.19731 

2.08043 

2.30413 

2.19294 

2.41670 

2.31094 

2-53532 

2.43471 

Q-55 

1.92509 

2.12185 

2.01814 

2.23725 

2.11624 

2.35824 

2.21964 

2.48513 

2.32858 

2.61823 

0.6 

1.74231 

2.25749 

1.82653 

2.38027 

i-9I53i 

2.50900 

2.00889 

2.64400 

2.10749 

2.78561 

0.65 

1.54^78 

2.37922 

1.62365 

2.50862 

1.70258 

2.64429 

1.78576 

2.78657 

r.87341 

2.93582 

0.7 

I-3457I 

2.48627 

1.41076 

2.62149 

1-47934 

2.76327 

1-55162 

2.91196 

1.62777 

3.06792 

o-75 

I-I3435 

2.57800 

1.18918 

2.71821 

1.24698 

2.86522 

1.30791 

3.01939 

1.37210 

3.18111 

0.8 

0.91599 

2-65384 

0.96026 

2.79817 

1.00694 

2.94950 

1.05614 

3.10821 

1.10797 

3.27468 

0.85 

0.69198 

2.71331 

0.72542 

2.86088 

0.76069 

3.01560 

0.79785 

3.17787 

0.83701 

3-34807 

0.9 

0.46370 

2.75606 

0.48612 

2.90595 

0.50975 

3.06311 

0.53465 

3-22793 

0.56089 

3.40081 

o-95 

0.23257 

2.78181 

0.24381 

2.93310 

0.25566 

3-09173 

0.26815 

3.25809 

0.28131 

3.43259 

I.O 

o.oo 

2.79041 

o.oo 

2.94217 

o.oo 

3.10129 

o.oo 

3.26816 

0.00 

3-44321 

1.05 

0.23257 

2.78181 

0.24381 

2.93310 

0.25566 

2.09173 

0.26815 

3.25809 

0.28131 

3-43259 

i.i 

0.46370 

2.75606 

0.48612 

2-90595 

0.50975 

3.06311 

0.53465 

3-22793 

0.56089 

3.40081 

i-i5 

0.69198 

2.71331 

0.72542 

2.86088 

0.76069 

3.01560 

0.79785 

3.17787 

0.83701 

3.34807 

1.2 

0.91599 

2.65384 

0.96026 

2.79817 

1.00694 

2.94950 

1.05614 

3.10821 

1.10797 

3-27468 

1.25 

.13435 

2.57800 

1.18918 

2.71821 

1.24698 

2.86522 

1.30791 

3-OI939 

1.37210 

3.18111 

i-3 

.34571 

2.48627 

1.41076 

2.62149 

1.47934 

2.76327 

1.55162 

2.91196 

1.62777 

3.06792 

i-35 

.54878 

2.37922 

1.62365 

2.50862 

1.70258 

2.64429 

1.78576 

2.78657 

1.87341 

2.93582 

1.4 

.74231 

2.25749 

1.82653 

2.38027 

1-91531 

2.50900 

2.00889 

2.64400 

2.10749 

2.78561 

i-4S 

.92509 

2.12185 

2.01814 

2.23725 

2.11624 

2.35824 

2.21964 

2.48513 

2.32858 

2.61823 

i-5 

2.09600 

1.97312 

2.19731 

2.08043 

2.30413 

2.19294 

2.41670 

2.31094 

2.53532 

2.43471 

i-SS 

2.25399 

1.81223 

2.36294 

1.91079 

2.47780 

2.01413 

2.59887 

2.12250 

2.72642 

2.23618 

1.6 

2.39808 

1.64016 

2.51400 

1.72937 

2.63620 

1.82289 

2.76501 

1.92098 

2.90071 

2.02387 

1.65 

2.52739 

1-45799 

2.64956 

1-53728 

2.77835 

1.62042 

2.91409 

1.90761 

3.05713 

1.79907 

1-7 

2.64111 

1.26682 

2.76878 

1-33572 

2.90337 

1.40796 

3.04522 

1-48372 

3.19469 

1.56318 

i-75 

2.73855 

1.06784 

2.87093 

1.12592 

3.01049 

1.  18681 

3.15757 

1.25067 

3.31255 

1.31766 

1.8 

2.81911 

0.86229 

2.95538 

0.90918 

3.09904 

0.95835 

3-25045 

1.00992 

3.41000 

1.06401 

1.85 

2.88229 

0.65141 

3.02161 

0.68684 

3.16850 

0.72398 

3-32330 

0.76294 

3.48641 

0.80380 

1.9 

2.92769 

0.43652 

3.06921 

0.46026 

3.21841 

0.48627 

3.37565 

0.51125 

3.54134 

0.53864 

i-95 

2.95505 

0.21893 

3.09789 

0.23084 

3.24848 

0.24332 

3.40719 

0.25642 

3.57443 

0.27015 

2.O  2.96419     O.OO 


3.10747   o.oo 


3.25853   o.oo 


3.41773    o.oo 


3.58548   o.oo 


Note.  Negative  quantities  are  in  heavy  type. 

Examples,    cosh  (1.8  +  «  o.  2)  =  2.95538  +  i  0.90918. 
cosh  (1.8  +  i  2.0)  =  —3.10747  +  »o. 


[65] 


TABLE  VIII.     HYPERBOLIC  COSINES,     cosh  (x  +  iq)  =  u  +  iv.     CONTINUED 


9 

x  =  2.0 

x  =  2.05 

X  =  2.1 

x  =  2.15 

X  =  2.2 

o 
0.05 

O.I 

0.15 

0.2 

3.76220  o.oo 
3.75059  0.28456 
3.71587  0.56737 
3.65825  0.84667 
3.57806  1.12076 

3.94832  o.oo 

3.93615  0.29968 
3.89971  0.59751 
3.83923  0.89166 
3.75507  1.18032 

4.14431  o.oo 
4.13154  0.31555 
4.09329  0.62916 
4.02981  0.93888 
3.94148  1.24282 

4.35067  o.oo 

4.33726  0.33221 
4.29711  0.66237 
4.23046  0.98845 
4-13773  1-30844 

4.56791  o.oo 

4.55382  0.34970 

4.51167  0.69724 
4.44170  1.04049 
4-34433  1-37732 

0.25 

0-35 
0.4 

0-45 

3.47581  1.38794 
3.35214  1.64656 
3.20780  1.89503 
3.04368  2.13182 
2.86080  2.35546 

3.64777  1.46169 
3.51798  1.73405 
3.36649  1.99573 
3.19426  2.24509 
3.00232  2.48062 

3.82885  1.53910 
3.69261  1.82589 
3.53361  2.10142 
3.35283  2.36399 
3.15137  2.61199 

4.01950  1.62035 
3.87647  1.92228 
3.70956  2.21236 
3.51977  2.48879 
3.30828  2.74988 

4.22019  1.70566 
4.07003  2.02349 
3.89478  2.32883 
3.69552  2.61982 
3-47347  2.89466 

0-5 
0-55 

0.6 
0.65 
0-7 

2.66027  2.56458 
2-44335  2.75789 
2.21136  2.93420 

I-96574  3-09241 
1.70800  3.23156 

2.79188  2.70085 

2.56423  2.90443 

2.32076  3.09011 
2.06299  3.23673 
1.79250  3.40327 

2.93048  2.84389 
2.69152  3.05825 
2-43597  3-25376 
2.16540  3.42920 
1.88148  3-5835I 

3.07639  2.99402 
2.82553  3.21970 
2.55726  3-42553 

2.27322  3.61024 

1.97516  3-77269 

3.23000  3.15165 
2.96661  3.38921 
2.68495  3.60588 
2.38672  3.80031 
2.07379  3.97I3I 

0.75 
0.8 
0.85 
0.9 
0.95 

1-43973  3-35078 
1.16258  3-44935 
0.87827  3.52666 
0-58854  3.58221 
0.29518  3.61568 

1.51096  3.52883 

I.22OIO  3.63264 
0.92172  3.71404 
0.61765  3.77256 
0.30978  3.80781 

1-58596  3-7I57I 
1.28066  3.82501 

0.96747  3-9I074 
0.64831  3-97235 
0.32516  4.00946 

1.66493  3-91188 

1-34443  4-02695 
1.01564  4.11720 
0.68059  4.18206 
0.34135  4-22113 

1.74806  4.11783 
1.41156  4.23896 
1.06636  4-33396 
0.71458  4.40223 
0.35839  4-44337 

I.O 

1.05 
i.i 

1.2 

o.oo    3.62686 
0.29518  3.61568 
0.58854  3.58221 
0.87827  3.52666 
1.16258  3-44935 

o.oo    3.81958 

0.30978  3.80781 
0.61765  3.77256 
0.92172  3.71404 
1.  22010  3.63264 

o.oo    4.02186 
0.32516  4.00946 

0.64831  3-97235 
0.96747  3-91074 

1.28066  3.82501 

o.oo  4.23419 
0.34135  4-22113 

0.68059  4.18206 
1.01564  4.11720 
1.34443  4-02695 

0.00      4-457II 

0.35839  4-44337 
0.71458  4.40223 
1.06636  4-33396 
1.41156  4.23896 

1.25 

i-3S 
1.4 

MS 

1-43973  3-35078 
1.70800  3.23156 

I-96574  3-09241 
2.21136  2.93420 
2.44335  2.75789 

I.5I096  3.52883 
1.79250  3-40327 
2.06299  3.23673 
2.32O76  3.OOOII 
2.56423  2.90443 

1.58596  3-71571 
1.88148  3.58351 

2.16540  3.42920 
2.43597  3-25376 
2.69152  3.05825 

1.66493  3.91188 
i.975i6  3.77269 
2.27322  3.61024 
2.55726  3.42553 
2.82553  3-2I970 

1.74806  4.11783 
2.07379  3-97I3I 
2.38672  3.80031 
2.68495  3-60588 
2.96661  3.38921 

i-SS 
.6 

.65 
•7 

2.66027  2.56458 
2.86080  2.35546 
3.04368  2.13182 
3.20780  1.89503 
3.35214  1.64656 

2.79188  2.70085 
3.OO232  2.48062 
3.19426  2.24509 
3.36649  1.99573 
3.51798  L73405 

2.93048  2.84389 
3.15137  2.61199 
3.35283  2.36399 
3.53361  2.10142 
3.69261  1.82589 

3.07639  2.99402 
3.30828  2.74988 
3.51977  2.48879 
3.70956  2.21236 
3.87647  1.92228 

3.23000  3.15165 
3-47347  2.89466 
3.69552  2.61982 
3.89478  2.32883 
4.07003  2.02349 

•75 
.8 

-85 
1.9 

3.47581  1.38794 
3.57806  1.12076 
3.65825  0.84667 
3.71587  0.56737 
3.75059  0.28456 

3.64777  1.46169 
3-75507  I.I8032 
3.83923  0.89166 
3.89971  0.59751 
3.93615  0.29968 

3.82885  1.53910 

3.94148  1.24282 
4.02981  0.93888 
4.09329  0.62916 

4.13154  0.31555 

4.01950  1.62035 
4-13773  1-30844 
4.23046  0.98845 
4.29711  0.66237 
4.33726  0.33221 

4.22019  1.70566 
4-34433  1-37732 
4.44170  1.04049 
4.51167  0.69724 
4.55382  0.34970 

2.O 

3.76220  o.oo 

3.94832  o.oo 

4.14431  o.oo 

4.35067  o.oo 

4.56791  o.oo 

Note. 

Negative  quantities  are  in  heavy  type.              • 

Examples,  cosh  (2.1  +  *  0.8)  =  1.28066  +  i  3.82501. 

cosh  (2.2  +  i  1.25)  =  —1.74806  +  i  4.11783. 


[66] 


TABLE  VIII.     HYPERBOLIC  COSINES,     cosh  (*  +  iq)  =  u  +  iv.     CONTINUED 


2.25 


x  =  2.3 


x  =  2.35 


2.4 


x  =  2.45 


o 
0.05 

O.I 

0.15 

O.2 

4.79657  o.oo 

4.78178  0.36807 
4-737SI  0.73386 
4.66404  1.09513 
4.56l8l  1.44965 

5.03722  o.oo 

5.02169  0.38735 
4.97521  0.77231 
4.89805  1.15251 
4.79058  1.52560 

5.29047  o.oo 
5.27416  0.40760 
5.22533  0.81269 
5.14429  1.21277 
5.03153  1.60537 

5.55695  o.oo 

5.53981  0.42888 
5.48853  0.85511 
5.40341  1.27607 

5.28496  1.68916 

5.83732  o.oo 

5.81933  0.45122 
5.76545  0.89966 
5.67603  1.34255 
5.55162  1.77716 

0.25 

o-3 
o-3S 
0.4 

0-45 

443145  1.79523 
4.27377  2.12975 
4.08975  2.45II3 
3.88050  2.75740 
3-64734  3-04677 

4.65378  1.88930 
4.48820  2.24134 

4.29494  2.57956 

4.07520  2.90188 
3.83034  3.20630 

4.88776  1.98808 
4.71384  2.35853 
4.51087  2.71443 
4.28008  3.05360 
4.02200  3.37395 

5.13394  2.09184 

4.95127  2.48162 
4.73808  2.85610 
4.49567  3.21297 
4.22554  3.55003 

5.39298  2.20082 
5.20109  2.61091 

4-97713  3-00490 
4.72249  3.38036 

443873  3-73499 

o-5 
0-55 
0.6 
0.65 
0.7 

3.39168  3.3I7I6 
3.II5I2  3.567I-9 
2.81935  3.79523 
2.50620  3.99987 
2.17760  4.17986 

3.56185  3.49096 
3.27141  3.75410 

2.96081  3.99409 
2.63194  4.20946 

2.28685  4.39887 

3-74093  3.67351 
3.43588  3.95038 

3.10966  4.20292 

2.76426  4.42955 
2.40182  4.62887 

3-92935  3-86521 
3.60894  4-15656 

3.26629  4.42228 

2.90349  4.66073 
2.52280  4.87045 

4.12761  4.06659 
3.79104  4-373" 
3.43109  4.65268 
3.04999  4.90356 
2.65009  5.12420 

0-75 
0.8 
0.85 
0.9 
o-95 

I-83557  4-33407 
1.48222  4.46157 

I.II974  4-56I5S 
0.75035  4-6334I 
0.37633  4.6767I 

1.92766  4.56116 

1.55659  4-69533 
1.17592  4.80056 
0.78799  4.87618 

0.39522  4-92174 

2.02457  4.79965 
1.63485  4.94083 
1.23504  5.05156 
0.82761  5.13114 
0.41509  5.17909 

2.12655  5.05014 
1.71719  5.19869 
1.29724  5.31521 
0.86930  5.39893 
0.43599  5-44938 

2-23385  5-3I325 
1.80383  5-46955 
1.36269  5.59213 
0.91316  5.68022 
0-45799  5-73330 

I.O 

1-05 
i.i 
"5 

1.2 

o.oo  4.69117 

0.37633  4-67671 
0.75035  4-63341 
1.11974  4.56155 

1.48222  4.46157 

o.oo  4.93696 

0.39522  4.92174 
0.78799  4.87618 
1.17592  4.80056 

1.55659  4.69533 

o.oo    5-I95Io 
0.41509  5.17909 
0.82761  5.13114 
1.23504  5.05156 
1.63485  4-94083 

o.oo  5.46623 

0.43509  5.44938 
0.86930  5.39893 
1.29724  5.31521 
1.71719  5.19869 

o.oo  5-75103 

0.45799  5-73330 

0.91316  5.68022 

1.36269  5.59213 
1.80383  5-46955 

1.25 

i-3 
1-35 
1.4 

1-45 

1.83557  4.33407 

2.17760  4.17986 

2.50620  3.99987 

2.81935  3.79523 
3.11512  3.56719 

1.92766  4.56116 
2.28685  4.39887 
2.63194  4.20946 
2.96081  3.99409 

3.27141  3.75410 

2.02457  4.79965 
2.40182  4.62887 
2.76426  4-42955 
3.10966  4.20292 
3.43588  3.95038 

2.12655  5.05014 
2.52280  4.87045 

2.90349  4.66073 

3.26629  4.42228 

3.60894  4.15656 

2.23385  5-3r325 
2.65009  5.12420 
3.04999  4-90356 
3.43109  4.65268 
3.79104  4.37311 

i-5 
i-55 
1.6 

1.65 
i-7 

3.39168  3.31716 

3.64734  3-04677 
3.88050  2.75740 
4.08975  2.45113 
4.27377  2.12975 

3.56185  3.49096 
3.83034  3.20630 
4.07520  2.90188 

4.29494  2.57956 

4.48820  2.24134 

3.74093  3-6735I 
4.02290  3-37395 
4.28008  3.05360 
4.51087  2.71443 
4.71384  2.35853 

3.92935  3-86521 
4.22554  3-55003 
4.49567  3-21297 

4.73808  2.85610 
4.95127  2.48162 

4.12761  4.06659 

4.43873  3-73499 
4.72249  3.38036 
4-97713  3-00490 
5.20109  2.61091 

i-75 
1.8 

i.«5 
1.9 

i-95 

4.43145  1.79523 
4.56181  1.44965 
4.66404  1.09513 
4.73751  0.73386 
4.78178  0.36807 

4.65378  1.88930 
4.79058  1.52560 
4.89805  1.15251 
4.97521  0.77231 
5.02169  0.38735 

4.88776  1.98808 
5.03153  1-60537 
5.14429  1.21277 
5-22533  0.81269 
5.27416  0.40760 

5.13394  2.09184 

5.28496  1.68916 

5.40341  1.27607 
5.48853  0.85511 
5.53981  0.42888 

5.39298  2.20082 
5.55162  1.77716 
5.67603  1.34255 
5.76545  0.89966 
5.81933  0.45122 

2.O 

4.79657  o.oo 

5.03722  o.oo 

5.29047  o.oo 

5.55<595  o.oo 

5.83732  o.oo 

Note. 


Negative  quantities  are  in  heavy  type. 


Examples,     cosh  (2.4  +  10.4)  =  4-4956?  +  *  3-21297- 
cosh  (2.4  +  i  1.5)  =  —3-92935  +  *  3-86521 


[67] 


TABLE  VIII.     HYPERBOLIC   COSINES,     cosh  (x  +  iq)  =  u  +  iv.     CONTINUED 


q 

x  =  2.5 

x  =  2.55 

X  =  2.6 

x  =  2.65 

x  =  2.7 

0 

0.05 

O.I 

0.15 

0.2 

6.13229  o.oo 

6.II339  0.47469 
6.05680  0.94646 
5.96287  I.4I239 
5.83215  1.86961 

6.44259  o.oo 

6.42273  0.49935 
6.36327  0.99563 
6.26458  1.48577 

6.12727  1.96674 

6.76901  o.oo 
6.74814  0.52526 
6.68567  1.04729 

6.58199  1.56285 
6.43770  2.06878 

7.11234  o.oo 

7.09042  0.55249 

7.02478  1.10156 

6.91583  1.64385 

6.76424  2.17600 

7-47347  o.oo 
7.45043  0.58109 
7.38146  1.15859 
7.26697  1.72896 
7.10769  2.28866 

0.25 
0-3 

0-35 
0.4 

0-45 

5.66550  2.3I53I 
5.46392  2.74674 
5.22864  3.l6l22 
4.96113  3.55622 
4.66304  3.92929 

5.95218  2.43559 
5.74039  2.88943 
5.49321  3.32545 
5.21217  3.74097 
4.89898  4-13342 

6.25374  2.56196 
6.03123  3.03934 

5-77153  3-49799 
5.47624  3.93506 
5.14719  4-34788 

6.57095  2.69474 

6.33714  3-I9686 
6.06427  3.67927 
5.75401  4-13900 
5.40827  4-57321 

6.90458  2.83425 
6.65891  3.36237 
6.37218  3.86976 
6.04606  4-35329 
5.68287  4.80998 

0-5 

0-55 
0.6 
0.65 
0.7 

4.33619  4.27814 
3.98260  4.60062 
3.60448  4-89472 
3.204II  5.15865 
2.78401  5.39077 

4.55560  4.50039 
4.18413  4.83961 
3.78686  5.14900 
3.36624  5.42664 
2.92488  5.67082 

4.78641  4-73389 
4.39612  5.09071 
3.97872  5-41615 
3.53680  5.70819 
3.07306  5.96505 

5.02919  4.97923 
4.61910  5.35454 
4.18053  5.69685 
3.71619  6.00403 
3.22894  6.27419 

5.28454  5.23702 
4-85363  5-63177 
4.39279  5.99I79 
3.90488  6.31488 
3.39288  6.59903 

0-75 
0.8 
0.85 
0.9 
0-95 

2.34673  5.S8966 
1.89498  5.75408 
I.43I5S  5.88304 
0-9593°  5-97572 
0.48113  6.03155 

2.46547  5.88004 
1.99087  6.05301 
1.50399  6.18866 
1.00784  6.28615 
0.50548  6.34489 

2.59039  6.18512 
2.09174  6.36706 
1.58019  6.50976 
1.05891  6.61231 
0.53109  6.67409 

2.72178  6.50567 
2.19784  6.69705 
1.66034  6.84713 
1.11262  6.95500 
0-55803  7.01998 

2.85997  6.84249 

2-30943  7-04377 
1.74465  7.20163 
1.16911  7.31508 
0.58636  7.38343 

I.O 

1.05 
i.i 
i.iS 

1.2 

o.oo  6.05020 
0.48113  6.03155 
0.95930  5-97572 
1.43155  5.88304 
1.89498  5.75408 

o.oo  6.36451 

0.50548  6.34489 

1.00784  6.28615 
1.50399  6.18866 
1.99087  6.05301 

o.oo    6.69473 
0.53109  6.67409 
1.05891  6.61231 
1.58019  6.50976 
2.09174  6.36706 

o.oo  7.04169 
0.55803  7.01998 
1.11262  6.95500 
1.66034  6.84713 
2.19784  6.69705 

o.oo    7.40626 

0.58636  7.38343 
1.16911  7.31508 
1.74465  7.20163 
2.30943  7-04377 

1.25 

i-3 
i-35 
1.4 

1-45 

2.34673  5.58966 
2.78401  5.39077 
3.20411  5.15865 

3.60448  4.89472 
3.98260  4.60062 

2.46547  5.88004 

2.92488  5.67082 
3.36624  5.42664 
3.78686  5.14900 
4.18413  4.83961 

2.59039  6.18512 
3.07306  5.96505 
3.53680  5.70819 
3.97872  5.41615 
4.39612  5.09071 

2.72178  6.50567 
3.22894  6.27419 
3.71619  6.00403 

4.18053  5.69685 
4.61910  5.35454 

2.85997  6.84249 
3.39288  6.59903 
3.90488  6.31488 
4-39279  5-99179 
4-85363  5-63177 

i-5 
i-55 
1.6 

1.65 
i-7 

4.33619  4.27814 
4.66304  3.92929 
4.96113  3.55622 
5.22864  3.16122 
5.46392  2.74674 

4.55560  4.50039 
4.89898  4.13342 
5.21217  3.74097 
5.49321  3.32545 
5.74039  2.88943 

4.78641  4.73389 
5.14719  4.34788 
5.47624  3.93506 

5-77153  3-49799 
6.03123  3.03934 

5.02919  4.97923 

5.40827  4-57321 
5.75401  4.13900 
6.06427  3.67927 
6.33714  3-19686 

5.28454  5.23702 
5.68287  4.80998 
6.04606  4-35329 
6.37218  3.86976 
6.65891  3.36237 

i-75 
1.8 
1.85 
1.9 

1-95 

5.66550  2.31531 
5.83215  1.86961 
5.96287  1.41239 

6.05680  0.94646 
6.1  1339  0.47469 

5.95218  2.43559 

6.12727  1.96674 

6.26458  1.48577 
6.36327  0.99563 
6.42273  0.49935 

6.25374  2.56196 
6.43770  2.06878 
6.58199  1.56285 
6.68567  1.04729 
6.74814  0.52526 

6.57095  2.69474 
6.76424  2.17600 
6.91583  1.64385 
7.02478  1.10156 
7.09042  0.55249 

6.90458  2.83425 
7.10769  2.28866 
7.26697  1.72896 

7.38146  1.15859 
7.45043  0.58109 

2.0 

6.13229  0.00 

6.44259  o.oo 

6.76901  o.oo 

7.11234  o.oo 

7-47347  o.oo 

Note.  Negative  quantities  are  in  heavy  type. 

Examples,    cosh  (2.7  +  *  i.oo)  =  o  +  i  7.40626. 

cosh  (2.6  +  j  1.2)  =  —2.09174  +  i 6.36706. 


[68] 


TABLE  VIII.     HYPERBOLIC   COSINES,     cosh  (x  +  iq)  =  u  +  iv.     CONTINUED 


x  =  2.75 


x  =  2.80 


x  =  2.85 


x  =  2.90 


x  =  2.95 


o 
0.05 

O.I 

0.15 

O.2 

7.85328  o.oo 
7.82907  0.61115 

7.75659  1.21852 
7.63629  1.81839 

7.46891  2.40704 

8.25273  o.oo 
8.22728  0.64273 
8.15112  1.28150 

8.02470  1.91237 
7.84881  2.53144 

8.67281  o.oo 
8.64608  0.67592 
8.56604  1.34768 

8.43318  2.OIII2 
8.24834  2.66217 

9.11458  o.oo 

9.08649  0.71081 

9.00237  1.41723 
8.86275  2.11491 
8.66850  2.79956 

9.57915  o.oo 

9.54962  0.74747 

9.46121  1.49032 

9.31447  2.22399 
9.11031  2.94395 

0.25 
0-3 

o-35 
0.4 

0-45 

7.25548  2.98086 
6.99732  3.53629 

6.69602  4.06993 

6.35344  4-57847 
5.97168  5.05878 

7.62452  3.13491 
7-35323  3-71905 

7.03661  4.28026 
6.67660  4.81509 
6.27542  5.32022 

8.01263  3.29681 
7.72754  3-9HI2 
7-39479  4-50I3I 
7.01646  5.06375 
6.59486  5.59498 

8.42078  3.46694 

8.12115  4.11295 
7.77146  4-73361 
7-37385  5-32508 
6.93078  5.88371 

8.84998  3.64575 
8.53508  4.32508 

8.16757  4-97774 
7.74969  5.63048 
7.28404  6.18717 

o-S 
0-55 
0.6 
0.65 
0.7 

5-55311  5-50790 
5.10030  5.92307 

4.61604  6.30172 

4.10333  6.64151 
3.56531  6.94036 

5-83556  5-79256 
5.35972  6.22918 
4.85083  6.62740 
4.31204  6.98476 
3.74666  7.29905 

6.13261  6.09170 
5.63254  6.55088 
5.09775  6.96966 

4.53I53  7-34547 
3.93738  7.67599 

6.44498  6.40608 

5.91945  6.88894 
5-35742  7-32934 
4.76236  7.72455 
4.13793  8.07213 

6.77348  6.73647 
6.22116  7.24424 
5.63048  7.70735 
5.00509  8.12294 
4.34884  8.48845 

0-75 
0.8 
0.85 
0.9 
o-95 

3.00532  7.19642 

2.42680  7.40811 

1-83331  7-57413 
1.22852  7.69345 

0.61616  7.76534 

3.15818  7.56834 
2.55023  7-79098 
1.92656  7-96557 

1.29101  8.09106 
0.64750  8.16666 

3.31894  7-959I9 
2.68005  8.19332 
2.02463  8.37694 
I-35673  8.50891 
0.68046  8.58841 

3.48800  8.36994 
2.81656  8.61616 
2.12776  8.80924 
1.42583  8.94802 
0.71512  9.03163 

3.66578  8.80162 
2.96012  9.06053 
2.23621  9.26358 
1.49851  9.40952 
0-75*57  9-49744 

I.O 

1.05 
i.i 
i-iS 

1.2 

o.oo  7-78935 
0.61616  7-76534 
1.22852  7.69345 

1.83331  7-574I3 
2.42680  7.40811 

o.oo    8.19192 
0.64750  8.16666 
1.29101  8.09106 

1.92656  7-96557 
2.55023  7-79098 

o.oo    8.61497 
0.68046  8.58841 
1-35673  8.50891 
2.02463  8.37694 
2.68005  8.19332 

o.oo  9-05956 
0.71512  9.03163 
1.42583  8.94802 
2.12776  8.80924 
2.81656  8.61616 

o.oo  9.52681 
0.75157  9-49744 
1.49851  9.40952 
2.23621  9.26358 
2.96012  9.06053 

1-25 

i-3 
1-35 
1.4 

1-45 

3.00532  7.19642 
3-5653I  6.94036 
4.10333  6.64151 
4.61604  6.30172 
5.10030  5.92307 

3.15818  7-56834 
3.74666  7.29905 

4.31204  6.98476 
4.85083  6.62740 
5-35972  6.22918 

3.31894  7.95919 
3.93738  7-67599 
4.53I53  7-34547 
5.09775  6.96966 
5.63254  6.55088 

3.48800  8.36994 
4-13793  8.07213 
4.76236  7.72455 

5-35742  7-32934 
5.91945  6.88894 

3.66578  8.80162 
4.34884  8.48845 
5.00509  8.12294 
5.63048  7.70735 
6.22116  7.24424 

i-5 
1-55 
1.6 
1.65 
1-7 

5-553II  5.50790 
5.97168  5.05878 

6-35344  4-57847 
6.69602  4.06993 

6-99732  3-53629 

5.83556  5-79256 
6.27542  5.32022 
6.67660  4.81509 
7.03661  4.28026 

7.35323  3-71905 

6.13261  6.09170 
6.59486  5.59498 
7.01646  5.06375 
7.39479  4-50131 
7.72754  3-91112 

6.44498  6.40608 
6.93078  5.88371 
7-37385  5-32508 
7.77146  4.73361 
8.12115  4.11295 

6.77348  6.73647 
7.28404  6.18717 
7.74969  5-63048 
8.16757  4-97774 
8.53508  4.32508 

i-75 
1.8 
1.85 
1.9 
1-95 

7.25548  2.98086 
7.46891  2.40704 
7.63629  1.81839 
7.75659  1-21852 
7.82907  0.61115 

7.62452  3.1349! 
7.84881  2.53144 
8.02470  1.91237 
8.15112  1.28150 
8.22728  0.64273 

8.01263  3.29681 
8.24834  2.66217 

8.43318  2.0III2 
8.56604  1.34768 
8.64608  0.67592 

8.42078  3.46694 
8.66850  2.79956 
8.86275  2.11491 
9.00237  1.41723 
9.08649  0.71081 

8.84998  3.64575 
9.11031  2.94395 
9.31447  2.22399 
9.46121  1.49032 
9.54962  0.74747 

2.0 

7.85328  o.oo 

8.25273  o.oo 

8.67281  o.oo 

9.11458  o.oo 

9.579I5  0.00 

Note.  Negative  quantities  are  in  heavy  type. 

Examples,     cosh  (2.95  +  to)  =  9.57915  +  *'o. 

cosh  (2.8  +  i  1.2)  =  —2.55023  +  *  7-79098. 


[69] 


TABLE  VIII.     HYPERBOLIC   COSINES,     cosh  (*  +  ig)  =  u  +  iv.     CONTINUED 


x  =  3-0 


3-°S 


x  =  3.10 


=  3-iS 


=  3.20 


o 

10.06766 

o.oo 

10.58135 

o.oo 

11.12150 

o.oo 

11.68946 

o.oo 

12.28665 

0.00 

0.05 

10.03660 

0.78599 

10.54870 

0.82649 

11.08720 

0.86905 

11.65340 

0.91378 

12.24880 

0.96080 

O.I 

9-94371 

1.56714 

10.45110 

1.64788 

10.98460 

1.73274 

11.54550 

1.82193 

12.13540 

1.91568 

0.15 

9.78949 

2.33863 

10.28900 

2.45911 

10.81420 

2.58575 

11.36650 

2.71885 

11.94720 

2.85874 

O.2 

9-57492 

3-09569 

10.06350 

3.25518 

10.57720 

3.42281 

11.11730 

3.59900 

11.68530 

3.78419 

0.25 

9.30131 

3.83368 

9-77589 

4.03120 

10.27490 

4.23878 

10.79970 

4-45696 

11.35140 

4.68630 

o-3 

8.97035 

4.54806 

9-42805 

4-78233 

9-90933 

5.02860 

10.41540 

5-28745 

10.94750 

5-55952 

o.3S 

8.58409 

5-23433 

9.02209 

5.50400 

9.48264 

5-78743 

9.96690 

6.08533 

10.47610 

6.39846 

0.4 

8.14491 

5.88836 

8.56049 

6.19173 

8.99749 

6.51058 

9-45697 

6.84570 

9.94011 

.7-19795 

0-45 

7-655SI 

6.50609 

8.04612 

6.84128 

8.45686 

7.19358 

8.88873 

7-56387 

9.34284 

7-95306 

o-S 

7.11891 

7.08371 

7.48215 

7.44866 

7.86409 

7-83223 

8.26570 

8.23539 

8.68797 

8.65915 

o-SS 

6.53842 

7.61765 

6.87204 

8.OIOH 

7.22284 

8.42260 

7.59170 

8.85615 

7.97954 

9.31184 

0.6 

5.91762 

8.10463 

6.21956 

8.52218 

6.53705 

8.96104 

6.87089 

9.42230 

7.22191 

9.90712 

0.65 

5-26034 

8.54164 

5-52874 

8.98171 

5.81097 

9.44423 

6.10773 

9-93036 

6.41976 

10.44130 

0.7 

4.57062 

8.92599 

4.80383 

9.38586 

5.04906 

9.86919 

5.30690 

10.37720 

5.57802 

10.91120 

0-75 

3-85273 

9-25531 

4.04931 

9.73216 

4.25602 

10.23330 

4-47336 

10.76010 

4.70190 

11.31370 

0.8 

3.11108 

9-52757 

3.26982 

10.01840 

3-43673 

10.53430 

3.61224 

11.07660 

3.79678 

11.64650 

0.85 

2.35025 

9.74108 

2.47017 

10.24290 

2.59626 

10.77040 

2.72885 

11.32480 

2.86826 

11.90750 

0.9 

1-57493 

9-89454 

1.65529 

10.40430 

1-73979 

10.94010 

1.82863 

11.50320 

1.92205 

12.09510 

Q-9S 

o.  78990 

9.98699 

0.83020 

10.50150 

0.87258 

11.04230 

0.91714 

11.61070 

0.96400 

12.20810 

I.O 

0.00 

10.01787 

o.oo 

10.53399 

0.00 

11.07645 

o.oo 

11.64661 

o.oo 

12.24588 

1.05 

0.78990 

9.98699 

0.83020 

10.50150 

0.87258 

11.04230 

0.91714 

11.61070 

0.96400 

12.20810 

i.i 

i  -57493 

9-89454 

1.65529 

10.40430 

1.73979 

10.94010 

1.82863 

11.50320 

1.92205 

12.09510 

i-i5 

2.35025 

9.74108 

2.47017 

10.24290 

2.59626 

10.77040 

2.72885 

11.32480 

2.86826 

11.90750 

1.2 

3.11108 

9-52757 

3.26982 

10.01840 

3.43673 

10.53430 

3.61224 

11.07660 

3.79678 

11.64650 

1-25 

3-85273 

9-25531 

4.04931 

9.73216 

4.25602 

10.23330 

4-47336 

10.76010 

4.70190 

11.31370 

i-3 

4-57062 

8.92599 

4.80383 

9.38586 

5.04906 

9.86919 

5.30690 

10.37720 

5.57802 

10.91120 

1.35 

5-26034 

8.54164 

5.52874 

8.98171 

5.81097 

9-44423 

6.10773 

9.93036 

6.41976 

10.44130 

1.4 

5-91762 

8.10463 

6.21956 

8.52218 

6.53705 

8.96104 

6.87089 

9.42230 

7.22191 

9.90712 

i-45 

6.53842 

7.61765 

6.87204 

8.01011 

7.22284 

8.42260 

7.59170 

8.85615 

7-97954 

9.31184 

i-S 

7.11891 

7.08371 

7.48215 

7.44866 

7.86409 

7.83223 

8.26570 

8.23539 

8.68797 

8.65915 

i-5S 

7.65551 

6.50609 

8.04612 

6.84128 

8.45686 

7.19358 

8.88873 

7.56387 

9.34284 

7-95306 

1.6 

8.14491 

5.88836 

8.56049 

6.19173 

8.99749 

6.51058 

9.45697 

6.84570 

9.94011 

7-19795 

1.65 

8.58409 

5-23433 

9.02209 

5.50400 

9.48264 

5-78743 

9.96690 

6.08533 

10.47610 

6.39846 

i-7 

8.97035 

4.54806 

9.42805 

4-78233 

9.90933 

5.02860 

10.41540 

5.28745 

10.94750 

5-55952 

1-75 

9-30131 

3.83368 

9.77589 

4.03120 

10.27490 

4.23878 

10.79970 

4-45696 

11.35140 

4.68630 

1.8 

9-57492 

3-09569 

10.06350 

3-255i8 

10.57720 

3.42281 

11.11730 

3-59900 

11.68530 

3.78419 

1.85 

9.78949 

2.33863 

10.28900 

2.45911 

10.81420 

2.58575 

11.36650 

2.71885 

11.94720 

2.85874 

1.9 

9-94371 

1.56714 

10.45110 

1.64788 

10.98460 

1.73274 

11.54550 

1.82193 

12.13540 

1.91568 

1-95 

10.03660 

0.78599 

10.54870 

0.82649 

11.08720 

0.86905 

11.65340 

0.91378 

12.24880 

0.96080 

2.0 

10.06766 

0.00 

10.58135 

o.oo 

11.12150 

o.oo 

11.68946 

o.oo 

12.28665 

o.oo 

\ 

Note.     Negative  quantities  are  in  heavy  type. 

Examples,  cosh  (3.10  +  i  0.5)  = 

7.86409  + 

*  7.83223. 

cosh  (3.10  +  i  1.55)  =  -8.45686  +  i  7.19358. 


[7o] 


TABLE  VIII.     HYPERBOLIC   COSINES,     cosh  (*  +  iq)  =  u  +  iv.     CONTINUED 


=  3-25 


x  =  3.30 


=  3-35 


=  3-40 


3-45 


0 

12.91456 

o.oo 

I3-57476 

o.oo 

14.26891 

o.oo 

14.99874 

o.oo 

15-76607 

o.oo 

0.05 

12.87474 

I.OIO22 

I3-53290 

1.06217 

14.22498 

1.11677 

14.95250 

1.17417 

15.71746 

1.23450 

O.I 

12.75555 

2.OI422 

13.40764 

2.11780 

14.09323 

2.22666 

14.81410 

2.34110 

I5-57I96 

2.46139 

0.15 

12-55773 

3-00579 

13.19970 

3.16036 

13.87465 

3.32282 

14.58432 

3-49359 

I5-33045 

3.673" 

0.2 

12.28247 

3.97883 

12.91035 

4.18343 

I3-57054 

4.39850 

14.26465 

4.62455 

14.99442 

4.86217 

0.25 

11.93150 

4-92735 

I2.54U5 

5.18072 

13.18275 

5-44705 

13.85702 

5.72700 

14.56595 

6.02127 

o-3 

11.50695 

5.84548 

I2.O952O 

6.14670 

12.71369* 

6.46202 

13-36397 

6.79414 

14.04764 

7.14324 

o-35 

11.01147 

6.72758 

11.57440 

7.07352 

12.16623 

7.43715 

12.78852 

7-81938 

13.44278 

8.22116 

0.4 

10.44810 

7.56820 

IO.98222 

7.95737 

"•54379 

8.36643 

12.13423 

8.79642 

12.75501 

9.24840 

o.45 

9.82031 

8.36217 

10.32233 

8.79215 

10.85016 

9.24413 

11.40512 

9.71923 

11.98861 

10.21863 

0-5 

9.13197 

9.10456 

9.59881 

9.57273 

10.08964 

10.06483 

10.60570 

10.58212 

11.14830 

11.12585 

o-SS 

8.38733 

9.79082 

8.81610 

10.29428 

9.26691 

10.82349 

9.74090 

"-37975 

10.23924 

11.96447 

0.6 

7.59099 

10.41673 

7-97905 

10.95235 

8.38705 

11.51541 

8.81604 

12.10723 

9.26706 

12.72933 

0.65 

6.74784 

10.9/841 

7.09279 

11.54294 

7-45549 

12.13633 

7.83682 

12.76006 

8.23775 

13.41572 

0.7 

5.86309 

II.4724O 

6.16381 

12.06234 

6-47795 

12.68244 

6.80928 

I3-33423 

7-I5765 

14.01940 

0-75 

4.94218 

11.89566 

5-I9485 

12.50736 

5-46075 

13-15033 

5-73977 

13.82620 

6.03341 

14.53662 

0.8 

3.99082 

I2.2456O 

4.19483 

12.87530 

4.40933 

13-53717 

4.63487 

14.23291 

4.87198 

14.96423 

0.85 

3.01484 

I2.52OO2 

3.16897 

13.16388 

3-33IOI 

13.84054 

3-50I39 

14.55188 

3-68053 

15.29960 

0.9 

2.02028 

12.71726 

2.21356 

13.37120 

2.23215 

14.05851 

2.34632 

14.78111 

2.46636 

15.54061 

0-95 

1.01326 

12.83610 

1.06507 

13.49615 

I-II953 

14.18950 

1.17679 

14.91923 

1.23699 

15.68581 

I.O 

0.00 

12.87578 

o.oo 

13.53788 

o.oo 

14.23382 

0.00 

14.96536 

0.00 

15.73432 

1.05 

1.01326 

12.83610 

1.06507 

13.49615 

1.11953 

14.18950 

1.17679 

14.91923 

1.23699 

15.68581 

i.i 

2.02028 

12.71726 

2.12356 

13.37120 

2.23215 

14.05851 

2.34632 

14.78111 

2.46636 

15.54061 

i-iS 

3.01484 

I2.52OO2 

3-16897 

13.16388 

3.33101 

13.84054 

3.50139 

14.55188 

3.68053 

15.29960 

1.2 

3.99082 

12.24560 

4.19483 

12.87530 

4-40933 

13-53717 

4.63487 

14.23291 

4.87198 

14.96423 

1-25 

4.94218 

11.89566 

5.19485 

12.50736 

5-46075 

13-15033 

5-73977 

13.82620 

6.03341 

14.53662 

1-3 

5.86309 

II.4724O 

6.16381 

12.06234 

6.47795 

12.68244 

6.80928 

I3-33423 

7.15765 

14.01940 

•35 

6.74784 

10.9/841 

7.09279 

11.54294 

7-45549 

12.13633 

7.83682 

12.76006 

8.23775 

13.41572 

•4 

7-59099 

10.41673 

7.97905 

10.95235 

8.38705 

11-51541 

8.81604 

12.10723 

9.26706 

12.72933 

•45 

8.38733 

9.79082 

8.  81610 

10.29428 

9.26691 

10.82349 

9.74090 

"•37975 

10.23924 

11.96447 

•5 

9-13197 

9.10456 

9.59881 

9.57273 

10.08964 

10.06483 

10.60570 

10.58212 

11.14830 

11.12585 

•55 

9.82031 

8.36217 

10.32233 

8.79215 

10.85016 

9.24413 

11.40512 

9.71923 

11.98861 

10.21863 

.6 

10.44810 

7.56820 

10.98222 

7-95737 

11-54379 

8.36643 

12.13423 

8.79642 

12.75501 

9.24840 

1.65 

11.01147 

6.72758 

11.57440 

7-07352 

12.16623 

7.43715 

12.78852 

7-81938 

13.44278 

8.22116 

1-7 

11.50695 

5-84548 

12.09520 

6.14670 

12.71369 

6.46202 

13-3639? 

6.79414 

14.04764 

7.14324 

1-75 

11.93150 

4.92735 

12.54145 

5.18072 

13-18275 

5-44705 

13.85702 

5.72700 

14.56595 

6.02127 

1.8 

12.28247 

3.97883 

12.91035 

4-18343 

I3-57054 

4-39850 

14.26465 

4-62455 

14.99442 

4.86217 

1.85 

12.55773 

3-00579 

13.19970 

3.16036 

13-87465 

3.32282 

14.58432 

3-49359 

15.3304S 

3.67311 

1.9 

12.75555 

2.01422 

13.40764 

2.11780 

14.09323 

2.22666 

14.81410 

2.34110 

15-57196 

2.46139 

1.95 

12.87474 

I.OIO22 

I3-53290 

1.06217 

14.22498 

1.11677 

14.95250 

1.17417 

15.71746 

1.23450 

2.0      12.91456    o.oo 


13.57476     o.oo 


14.26891     o.oo 


14.99874    o.oo 


15.76607    o.oo 


Note.  Negative  quantities  are  in  heavy  type. 

Examples,     cosh  (3.45  +  i  0.05)  =  15.71746  +  i  1.23450. 
cosh  (3.25  +  i  1.95)  =  —12.87474  +  *  1.01022. 


TABLE  VIII.     HYPERBOLIC  COSINES,     cosh  (x  +  iq)  =  u  +  iv.     CONTINUED 


=  3-50 


=  3-55 


3.60 


x  =  3-65 


x  =  3.70 


o 

16.57282 

o.oo 

17.42102 

o.oo 

18.31278 

o.oo 

19.25033 

o.oo 

20.23601 

o.oo 

0.05 

16.52173 

1.29792 

17.36731 

1.36458 

18.25632 

1.43466 

19.19100 

1.50832 

20.17362 

1.58576 

O.I 

16.36878 

2.58783 

17.20653 

2.72075 

18.08732 

2.86047 

19.01331 

3-00735 

19.98688 

3.16174 

0.15 

16.11491 

3.86180 

16.93970 

4.06015 

17.80680 

4.26865 

18.71843 

4.48783 

19.67690 

4.71823 

O.2 

15.76170 

5.11195 

16.56840 

5.37451 

17.41650 

5.65052 

18.30814 

5-94065 

19.24560 

6.24563 

0.25 

i5-3"3o 

6.33059 

16.09492 

6.65574 

16.91880 

6.99754 

17.78497 

7.35683 

18.69565 

7-73453 

0-3 

14.76650 

7.51020 

15-52225 

7.89594 

16.31680 

8.30143 

17.1527.6 

8.72766 

18.03040 

9-17573 

o-3S 

14.13065 

8.64360 

14.85386 

9-08745 

15.61421 

9-55412 

16.41360 

10.04469 

17.25404 

10.56037 

0.4 

13.40770 

9.72361 

14.09390 

10.22293 

14-81535 

10.74789 

15-57383 

11.29972 

16.37127 

11.87990 

0-45 

12.60210 

10.74357 

13.24705 

11.29540 

13-92515 

11.87545 

14.63806 

12.48520 

15.38760 

13.12620 

o.S 

11.71820 

11.69740 

12.31852 

12.29820 

12.94910 

12.92978 

13.61203 

13-59365 

14.30902 

14-29155 

o-55 

10.76320 

12.57910 

11.31405 

13.22520 

11.89320 

13.90436 

12.50208 

14.61830 

13.14223 

15.36878 

0.6 

9.74126 

13.38330 

10.23982 

14.07066 

10.76400 

14.79326 

11.31505 

15-55281 

11.89444 

16.35127 

0.65 

8.65928 

14.10490 

9.10246 

14.82933 

9.56840 

15.59090 

10.05827 

16.39143 

10.57330 

17.23295 

0.7 

7-52391 

14.73960 

7.90898 

15.49665 

8.31383 

16.29246 

8.73946 

17.12900 

9.18696 

18.00840 

0-75 

6.34215 

15.28340 

6.66674 

16.06840 

7.00800 

16.89356 

7.36678 

17.76096 

7-74399 

18.67280 

0.8 

5.12129 

15-73300 

5-38339 

16.54105 

5.65896 

17.39050 

5.94868 

18.28342 

6.25327 

19.22208 

0.85 

3.86885 

16.08555 

4.06686 

16.91175 

4-27503 

17.78022 

4.49390 

18.69316 

4.72401 

19.65290 

0.9 

2.59256 

16.33896 

2.72525 

17.17817 

2.86475 

18.06033 

3.01142 

18.98765 

3.16561 

19.96246 

o-9S 

1.30029 

16.49163 

1.36684 

17-33870 

1.43680 

18.22900 

1.51036 

19.16506 

1.58770 

20.14900 

I.O 

o.oo 

16.54263 

0.00 

17.39230 

0.00 

18.28546 

o.oo 

19.22434 

0.00 

20.21129 

1.05 

1.30029 

16.49163 

1.36684 

17.33870 

1.43680 

18.22900 

1.51036 

19.16506 

1.58770 

20.14900 

i.i 

2.59256 

16.33896 

2.72525 

17.17817 

2.86475 

18.06033 

3.01142 

18.98765 

3.16561 

19.96246 

i-iS 

3.86885 

16.08555 

4.06686 

16.91175 

4.27503 

17.78022 

4.49390 

18.69316 

4.72401 

19.65290 

1.2 

5.12129 

15-73300 

5-38339 

16.54105 

5.65896 

17.39050 

5.94868 

18.28342 

6.25327 

19.22208 

I-2S 

6.34215 

15.28340 

6.66674 

16.06840 

7.00800 

16.89356 

7.36678 

17.76096 

7-74399 

18.67280 

•3 

7.52391 

14.73960 

7.90898 

15.49665 

8.31383 

16.29246 

8.73946 

17.12900 

9.18696 

18.00840 

•35 

8.65928 

14.10490 

9.10246 

14.82933 

9.56840 

15.59090 

10.05827 

16.39143 

io.5733o 

17.23295 

•4 

9.74126 

13-38330 

10.23982 

14.07066 

10.76400 

14.79326 

11.31505 

15-55281 

11.89444 

16.35127 

•45 

10.76320 

12.57910 

11.31405 

13.22520 

11.89320 

13.90436 

12.50208 

14.61830 

13.14223 

15.36878 

•5 

11.71820 

11.69740 

12.31852 

12.29820 

12.94910 

12.92978 

13.61203 

13-59365 

14.30902 

14-29155 

•55 

12.60210 

10.74357 

13.24705 

11.29540 

13.92515 

11-87545 

14.63806 

12.48520 

15.38760 

13.12620 

.6 

13.40770 

9.72361 

14.09390 

10.22293 

14.81535 

10.74789 

15.57383 

11.29972 

16.37127 

11.87990 

•65 

14.13065 

8.64360 

14.85386 

9.08745 

15.61421 

9-55412 

16.41360 

10.04469 

17.25404 

10.56037 

•7 

14.76650 

7.51020 

15.52225 

7-89594 

16.31680 

8.30143 

17.15216 

8.72766 

18.03040 

9-!7573 

i-7S 

15.31130 

6.33059 

16.09492 

6.65574 

16.91880 

6.99754 

17.78497 

7.35683 

18.69565 

7-73453 

1.8 

15.76170 

S-n^S 

16.56840 

5-37451 

17.41650 

5-65052 

18.30814 

5-94065 

19.24560 

6.24563 

1.85 

16.11491 

3.86180 

16.93970 

4.06015 

17.80680 

4.26865 

18.71843 

4.48783 

19.67690 

4.71823 

1.9 

16.36878 

2.58783 

17.20653 

2.72075 

18.08732 

2.86047 

19.01331 

3-00735 

19.98688 

3.16174 

i-95 

16.52173 

1.29792 

17.36731 

1.36458 

18.25632 

1.43466 

19.19100 

1.50832 

20.17362 

I-58576 

2.O 

16.57282 

0.00 

17.42102 

0.00 

18.31278 

o.oo 

19.25033 

o.oo 

20.23601 

o.oo 

Note.     Negative  quantities  are  in  heavy  type. 

Examples,  cosh  (3.50  +  *  0.70)  = 

7.52391  + 

*  14.73960 

. 

cosh  (3.60  +  *  1.55)  =  -13-92515  +  *  ".87545- 


[72] 


TABLE  VIII.     HYPERBOLIC  COSINES,     cosh  (*  +  iq)  =  u  +  iv.     CONTINUED 


*  =  3-75 


3-80 


3-85 


x  =  3.90 


3-95 


o 

21.27230 

o.oo 

22.36178 

o.oo 

23.50717 

o.oo 

24-71135 

o.oo 

25-97731 

o.oo 

0.05 

21.20670 

1.66716 

22.29283 

1-75273 

23.43470 

1.84268 

24.63516 

1.93724 

25.89724 

2.03665 

O.I 

21.01038 

3-32404 

22.08646 

3-49465 

23.21775 

3.67400 

24.40710 

3.86254 

25-65749 

4.06074 

0.15 

20.68452 

4.96043 

21.74391 

5-21503 

22.85766 

5-48267 

24.02856 

5.76402 

25-25957 

6.05979 

O.2 

20.23113 

6.56624 

21.26731 

6.90325 

22.35664 

7.25754 

23.50188 

7.62997 

24.70590 

8.02149 

0.25 

19.65301 

8.13156 

20.65958 

8.54892 

21.71778 

8.98766 

22.83030 

9.44887 

23.99991 

9-93373 

0-3 

i8.95373 

9.64674 

19.92448 

10.14188 

20.94503 

10.66233 

22.01797 

11.20952 

23-14597 

11.78474 

o-3S 

18.13760 

11.10246 

19.06655 

11.67230 

20.04315 

12.27134 

21.06988 

12.90106 

22.14931 

13-56305 

0.4 

17.20963 

12.48971 

18.09105 

13.13076 

19.01770 

13.80465 

19.99190 

14-51307 

21.01610 

I5-25776 

0.45 

16.17556 

13.79998 

1  7.00402 

14.50828 

17.87500 

15.25286 

18.79065 

16.03558 

19.75331 

16.85842 

o-S 

15.04177 

15.02516 

15.81216 

15-79634 

16.62208 

16.60702 

17-47355 

17.45924 

18.36873 

18.35512 

o-SS 

13.81524 

16.15770 

14.52281 

16.98701 

15.26668 

17.85880 

16.04874 

18.77526 

16.87098 

19.73867 

0.6 

12.50353 

17.19062 

13-14392 

18.07296 

13.81716 

19.00048 

14.52497 

19-97556 

15.26910 

21.00052 

0.65 

11.11473 

18.11756 

11.68400 

19.04746 

12.28246 

20.02501 

12.91164 

21.05272 

13-57315 

22.13290 

0.7 

9.65741 

18.93280 

10.15203 

19.90455 

10.67203 

20.92608 

11.21871 

21.99993 

11.79366 

23.12881 

o.7S 

8.14055 

19.63131 

8.55748 

20.63891 

8.99580 

21.69813 

9.45662 

22.81160 

9.94109 

23.98212 

0.8 

6.57350 

20.20879 

6.91017 

21.24603 

7.26411 

22.33640 

7.63623 

23.48262 

8.02744 

24.68760 

0.85 

4.96592 

20.66167 

5.22025 

21.72216 

5.48764 

22.83696 

5.76875 

24.00888 

6.06429 

25.24084 

0.9 

3.32772 

20.98716 

3-49815 

22.06437 

3-67733 

23.19673 

3-86571 

24.38710 

4.06375 

25.63849 

0-95 

1.66900 

21.18327 

1.75448 

22.27052 

1-84435 

23-41348 

1.93883 

24.61500 

2.03816 

25-87805 

I.O 

o.oo 

21.24878 

0.00 

22.33941 

o.oo 

23.48589 

0.00 

24.69110 

o.oo 

25.95806 

•05 

1.66900 

21.18327 

1.75448 

22.27052 

1.84435 

23.41348 

1.93883 

24.61500 

2.03816 

25.87805 

.1 

3.32772 

20.98716 

3.49815 

22.06437 

3.67733 

23.19673 

3.86571 

24.38710 

4.06375 

25.63849 

•IS 

4.96592 

20.66167 

5-22025 

21.72216 

5.48764 

22.83696 

5.76875 

24.00888 

6.06429 

25.24084 

.2 

6.57350 

20.20879 

6.91017 

21.24603 

7.26411 

22.33640 

7.63623 

23.48262 

8.02744 

24.68760 

•2S 

8.14055 

19-63131 

8.55748 

20.63891 

8.99580 

21.69813 

9.45662 

22.81160 

9.94109 

23.98212 

•3 

9.65741 

18.93280 

10.15203 

19.90455 

10.67203 

20.92608 

11.21871 

21.99993 

11.79366 

23.12881 

i-3S 

11.11473 

18.11756 

11.68400 

19.04746 

12.28246 

20.02501 

12.91164 

21.05272 

13.57315 

22.13290 

1.4 

12.50353 

17.19062 

13.14392 

18.07296 

13.81716 

19.00048 

14.52497 

19-97556 

15.26910 

21.00052 

MS 

13.81524 

16.15770 

14.52281 

16.98701 

15.26668 

17.85880 

16.04874 

18.77526 

16.87098 

19.73867 

*•$ 

15.04177 

15.02516 

15.81216 

15-79634 

16.62208 

16.60702 

17.47355 

17.45924 

18.36873 

18.35512 

x-55 

16.17556 

13.79998 

17.00402 

14.50828 

17.87500 

15.25286 

18.79065 

16.03558 

19.75331 

16.85842 

1.6 

17.20963 

12.48971 

18.09105 

13.13076 

19.01770 

13.80465 

19.99190 

14.51307 

21.01610 

15-25776 

1.65 

18.13760 

11.10246 

19.06655 

11.67230 

20.04315 

12.27134 

21.06988 

12.90106 

22.14931 

13-56305 

«.? 

18.95373 

9.64674 

19.92448 

10.14188 

20.94503 

10.66233 

22.01797 

11.20952 

23.14597 

11.78474 

1-75 

19.65301 

8.13156 

20.65958 

8.54892 

21.71778 

8.98766 

22.83030 

9.44887 

23.99991 

9-93373 

1.8 

20.23113 

6.56624 

21.26731 

6.90325 

22.35664 

7.25754 

23.50188 

7.62997 

24.70590 

8.02149 

1.85 

20.68452 

4.96043 

21.74391 

5-21503 

22.85766 

5.48267 

24.02856 

5.76402 

25.25957 

6.05979 

1.9 

21.01038 

3-32404 

22.08646 

3-49465 

23.21775 

3.67400 

24.40710 

3.86254 

25.65749 

4.06074 

i-9S 

21.20670 

1.66716 

22.29283 

1-75273 

23.43470 

1.84268 

24.635*6 

1.93724 

25.89724 

2.03665 

2.0 

21.27230 

o.oo 

22.36178 

o.oo 

23.50717 

o.oo 

24.71135 

o.oo 

35.97731 

0.00 

Note.     Negative  quantities  are  in  heavy  type. 

Examples,  cosh  (3.90  +  i  0.25)  = 

22.83030  +  i  9.44887. 

cosh  (3.75  +  i  1.25)  =  -8.14055  +  i  19.63131. 


[73] 


TABLE  IX.     HYPERBOLIC  TANGENTS,     tanh  (*  +  iq)  =  u  +  iv 


x  =  0.05 


x  =  0.15 


X  =  0.2 


0 

o.oo 

o.oo 

0.04996 

o.oo 

0.09967 

o.oo 

0.14889 

o.oo 

0.19738 

o.oo 

0.05 

o.oo 

0.07870 

0.05027 

0.07850 

0.10028 

0.07792 

0.14979 

0.07695 

0.19855 

0.07562 

O.I 

o.oo 

0.15838 

0.05121 

0.15798 

0.10214 

0.15677 

0.15254 

0.15479 

„  0.20213 

0.15207 

0.15 

o.oo 

0.24008 

0.05283 

0.23944 

0.10535 

0.23755 

0.15727 

0.23446 

0.20828 

0.23021 

0.2 

o.oo 

0.32492 

0.05522 

0.32402 

0.11008 

0.32136 

0.16422 

0.31698 

0.21732 

0.31098 

0.25 

o.oo 

0.41421 

0.05850 

0.41300 

0.11657 

0.40940 

0.17377 

0.40350 

0.22970 

0-39543 

o-3 

0.00 

0.50953 

0.06289 

0.50792 

0.12522 

0.50317 

0.18647 

0.49538 

0.24613 

0.48477 

0-35 

o.oo 

0.61280 

0.06865 

0.61070 

0.13659 

0.60446 

0.20310 

0.59427 

0.26758 

0.58044 

0.4 

o.oo 

0.72654 

0.07623 

0.72378 

0.15149 

0.71557 

0.22484 

0.70222 

0.29549 

0.68417 

0-45 

o.oo 

0.85408 

0.08624 

0.85040 

0.17113 

0.83951 

0.25339 

0.82186 

0.33192 

0.79813 

o.S 

o.oo 

I.OOOO 

0.09967 

0.99503 

0.19738 

0.98033 

0.29131 

0.95663 

0.37995 

0.92501 

0-55 

o.oo 

1.17085 

0.11804 

1-16395 

0.23313 

1.14365 

0.34258 

1.  11113 

0.44423 

1.06819 

0.6 

o.oo 

1.37638 

0.14392 

1.36649 

0.28315 

1-33754 

0.41357 

1.29164 

0.53203 

1-23185 

0.65 

o.oo 

1.63185 

0.18179 

1.61702 

0.35567 

1.57401 

0.51496 

1.50674 

0.65502 

1.42088 

0.7 

o.oo 

1.96261 

0.24007 

1.93900 

0.46575 

1.87150 

0.66202 

1.75880 

0.83268 

1.64005 

0-75 

o.oo 

2.41421 

0.33624 

2.37365 

0.64333 

2.25941 

0.90034 

2.09061 

1.09837 

1.89083 

0.8 

o.oo 

3.07768 

0.51109 

2.99911 

0.95397 

2.78504 

1.28858 

2.48723 

1.50982 

2.16055 

0.85 

o.oo 

4.16530 

0.87867 

3.98246 

1.56000 

3-51765 

1.97316 

2.94167 

2.16111 

2.38860 

0.9 

o.oo 

6.31375 

1.85674 

5.72808 

2.91746 

4.47780 

3.22989 

3-27758 

3-15925 

2.37676 

0-95 

o.oo 

12.70620 

5.79801 

9.05499 

6.21808 

4-83133 

5.28217 

2.71349 

4.39854 

1.67517 

I.O 

o.oo 

00 

20.01667 

o.oo 

10.03331 

o.oo 

6.71659 

o.oo 

5.06649 

o.oo 

1.05 

o.oo 

12.70620 

5.79801 

9.05499 

6.21808 

4.83133 

5.28217 

2.71349 

4.39854 

1.67517 

i.i 

o.oo 

6.31375 

1.85674 

5.72808 

2.91746 

4.47780 

3.22989 

3.27758 

3-15925 

2.37676 

i.iS 

o.oo 

4.16530 

0.87867 

3.98246 

1.56000 

3.51765 

1.97316 

2.94167 

2.16111 

2.38860 

1.2 

o.oo 

3.07768 

0.51109 

2.99911 

0.95397 

2.78504 

1.28858 

2.48723 

1.50982 

2.16055 

I-2S 

o.oo 

2.41421 

0.33624 

2.37365 

0.64333 

2.25941 

0.90034 

2.09061 

1.09837 

1.89083 

1-3 

o.oo 

1.96261 

0.24007 

1.93900 

0.46575 

1.87150 

0.66202 

1.75880 

0.83268 

1.64005 

i-3S 

o.oo 

1.63185 

0.18179 

1.61702 

0.35567 

1.57401 

0.51496 

1.50674 

0.65502 

1.42088 

1.4 

o.oo 

1.37638 

0.14392 

1.36649 

0.28315 

1-33754 

0.41357 

1.29164 

0.53203 

1.23185 

1-45 

0.00 

1.17085 

0.11804 

1.16395 

0.23313 

1.14365 

0.34258 

I.  IXII3 

0.44423 

1.06819 

i-S 

o.oo 

I.OOOO 

0.09967 

0.99503 

0.19738 

0.98033 

0.29131 

0.95663 

0-37995 

0.92501 

i-S5 

o.oo 

0.85408 

0.08624 

0.85040 

0.17113 

0.83951 

0.25339 

0.82186 

0.33192 

0.79813 

1.6 

o.oo 

0.72654 

0.07623 

0.72378 

0.15149 

0.71557 

0.22484 

0.70222 

0.29549 

0.68417 

1.65 

o.oo 

0.61280 

0.06865 

0.61070 

0.13659 

0.60446 

0.20310 

0.59427 

0.26758 

0.58044 

i-7 

o.oo 

0.50953 

0.06289 

0.50792 

0.12522 

0.50317 

0.18647 

0.49538 

0.24613 

0.48477 

i-75 

o.oo 

0.41421 

0.05850 

0.41300 

0.11657 

0.40940 

0.17377 

0.40350 

0.22970 

0.39543 

1.8 

o.oo 

0.32492 

0.05522 

0.32402 

0.11008 

0.32136 

0.16422 

0.31698 

0.21732 

0.31098 

1.85 

o.oo 

0.24008 

0.05283 

0.23944 

0.10535 

0.23755 

0.15727 

0.23446 

0.20828 

0.23021 

1.9 

o.oo 

0.15838 

0.05121 

0.15798 

0.10214 

0.15677 

0.15254 

0.15479 

0.20213 

0.15207 

i-9S 

o.oo 

0.07870 

0.05027 

0.07850 

0.10028 

0.07792 

0.14979 

0.07695 

0.19855 

0.07562 

o.oo     o.oo 


0.04996   o.oo 


0.09967   o.oo 


0.14889    o.oo 


0.19738   o.oo 


Note.  Negative  quantities  are  in  heavy  type. 

Examples,     tanh  (o  +  i  0.95)  =  o  4-  i  12.70620. 
tanh  (o  4-  i  1.45)  =  o  —  i  1.17085. 


[74] 


TABLE  IX.     HYPERBOLIC  TANGENTS,     tanh  (x  +  iq)  =  u  +  iv.     CONTINUED 


x  =  0.25 


=  0.3 


o-35 


x  =  0.4 


x  =  0.45 


o 
0.05 

O.I 

0.15 

O.2 

0.24492  o.oo 

0.24635  0.07395 

0.25069  0.14866 
0.25814  0.22490 
0.26907  0.30351 

0.29131  o.oo 
0.29296  0.07199 
0.29799  0.14464 
0.30660  0.21864 
0.31921  0.29471 

0.33638  o.oo 
0.33322  0.06975 
0.34384  0.14007 
0.35346  0.21153 
0.36750  0.28475 

0-37995  o.oo 
0.38196  0.06728 
0.38808  0.13503 
0-39853  0.20373 
0.41376  0.27384 

0.42190  o.oo 
0.42405  0.06462 
0.43056  0.12961 
0.44169  0.19534 
0.45784  0.26216 

0.25 
0-3 
0-35 
0.4 

0-45 

0.28402  0.38540 

0.30377  0.47162 
0.32947  0.56335 

0.36272  0.66200 
0.40582  0.76919 

0.33640  0.37362 
0.35903  0.45623 
0.38833  0.54348 

0.42600  0.63638 

0.47444  0.73604 

0.38658  0.36035 

0.41161  0.43898 

0.44383  0.52131 

0.48497  0.60802 
0-53739  0.69969 

0.43438  0.34585 
0.46130  0.42022 
0-49575  0.49737 
0.53941  0.57764 
0.59450  0.66116 

0.47964  0.33039 
0.50796  0.40033 
0-54397  0.47216 
0.58924  0.54593 
0.64580  0.62138 

o-S 
o-SS 
0.6 
0.65 
0.7 

0.46212  0.88682 

0.53655  1.01699 

0.63656  1.  16180 

0.77356  1.32269 

0.96528  1.49862 

0.53705  0.84355 

0.61869  0.95983 
0.72640  1.08513 
0.87037  1.  21810 

1.06521  1.35360 

0.60437  0.79671 
0.69041  0.89893 
0.80176  1.00519 

0.94684  1.  1  1  212 
1.13666  1.  21222 

0.66404  0.74770 
0.75200  0.83632 
0.86357  0.92478 
1.00528  1.00856 
1.18469  1.07919 

0.71630  0.69779 
0.80407  0.77365 
0.91321  0.84608 
1.04844  0.91003 
1.21438  0.95708 

o.7S 
0.8 
0.85 
0.9 
o-9S 

1.23914  1.68151 
1-63553  1-84484 

2.20223  1.91863 
2.95122  1.75011 

3.72316  1.11789 

1.33091  1.47820 

1.69121  1.56140 
2.16210  1.54177 
2.71602  1.31829 
3.21905  0.79096 

1.38412  I.29O2O 
1.70028  I.3I745 
2.08309  1.24667 
2.49443  I.OI6I3 
2.83603  0.58484 

1.40896  1.  12181 
1.68068  1.11235 
1.98936  1.01695 
2.29855  0.79978 
2.53928  0.44728 

1.41397  0.97400 
1.64487  0.94186 
1.89366  0.83750 
2.12957  0.64107 
2.30471  0.35122 

I.O 

1.05 
i.i 
i-iS 

1.2 

4.08299  o.oo 
3.72316  1.11789 

2.95122  1.75011 
2.20223  1.91863 

1-63553  1.84484 

3.43274  o.oo 
3.21905  0.79096 
2.71602  1.31829 
2.16210  1.54177 
1.69121  1.56140 

2.97287  o.oo 
2.83603  0.58484 
2.49443  1.01613 
2.08309  1.24667 
1.70028  1.31745 

2.63193  o.oo 
2.53928  0.44728 
2.29855  0.79978 
1.98936  .01695 
1.68068  .11235 

2.37024  o.oo 
2.30471  0.35122 

2.12957  0.64107 
1.89366  0.83750 
1.64487  0.94186 

1-25 

i-3 
1-35 
1.4 

1-45 

1.23914  1.68151 
0.96528  1.49862 
0.77356  1.32269 

0.63656  1.  16180 

0.53655  1-01699 

1.33091  1.47820 
1.06521  1.35360 
0.87037  1.  21810 
0.72640  1.08513 
0.61869  0.95983 

1.38412  1.29020 

1.13666   1.  21222 
0.94684  I.II2I2 
0.80176  I.OO5I9 
0.69041  0.89893 

1.40896  .12181 
1.18469   .07919 
1.00528  .00856 

0.86357  0.92478 
0.75200  0.83632 

1.41397  0.97400 

1.21438  0.95708 
1.04844  0.91003 
0.91321  0.84608 
0.80407  0.77365 

i-5 
i-55 
1.6 
1.65 
i-7 

0.46212  0.88682 
0.40582  0.76919 
0.36272  0.66200 

0.32947  0.56333 
0.30377  0.47162 

0.53705  0.84355 
0.47444  0.73604 

0.42600  0.63638 

0.38833  0.54348 
0-35903  0.45623 

O.6O437  0.79671 

0-53739  0.69969 
0.48497  0.60802 
0.44383  0.52131 
0.41161  0.43898 

0.66404  0.74770 
0.59450  0.66  1  16 
0.53941  0.57764 
0.49575  0.49737 
0.46130  0.42022 

0.71630  0.69779 
0.64580  0.62138 

0.58924  0.54593 
0.54397  0.47216 
0.50796  0.40033 

i-7S 
1.8 
1.85 
1.9 
J-95 

0.28402  0.38540 
0.26907  0.30351 
0.25814  0.22490 
0.25069  0.14866 

0.24635  0.07395 

0.33640  0.37362 
0.31921  0.29471 
0.30660  0.21864 
0.29799  0.14464 
0.29296  0.07199 

0.38658  0.36035 
0.36750  0.28475 
0.35346  0.21153 
0.34384  0.14007 
0.33822  0.06975 

0.43438  0.34585 
0.41376  0.27384 
0-39853  0.20373 
0.38808  0.13503 
0.38196  0.06728 

0.47964  0.33039 

0.45784  0.26216 
0.44169  0.19534 
0.43056  0.12961 
0.42405  0.06462 

2.O 

0.24492  o.oo 

0.29131  o.oo 

0-33638  o.oo 

0-37995  0.00 

0.42190  o.oo 

Note.  Negative  quantities  are  in  heavy  type. 

Examples,     tanh  (0.4  +  i  0.4)  =  0.53941  +  i  0.57764. 

tanh  (0.45  +  i  1.75)  =  0.47964  -  i  0.33039. 


[75] 


TABLE  IX.     HYPERBOLIC  TANGENTS,     tanh  (x  +  iq)  =  «  +  w.     CONTINUED 


o-5 


=  0.55 


x  =  0.6 


x  =  0.65 


=  0.7 


o 
0.05 

O.I 

0.15 

0.2 

0.46211  o.oo 
0.46436  0.06181 
0.47119  0.12390 
0.48281  0.18651 
0.49964  0.24990 

0.50052  o.oo 
0.50284  0.05889 
0.50987  0.11796 
0.52183  0.17737 
0.53910  0.23725 

0.53704  o.oo 
0.53941  0.05590 
0.54657  0.11189 
0.55872  0.16804 
0.57620  0.22437 

0.57167  o.oo 

0.57405  0.05288 

0.58125  0.10576 

0.59344  0.15863 

0.61094  0.21144 

0.60437  o-°° 
0.60674  0.04984 
0.61390  0.09962 
0.62602  0.14925 
0.64336  0.19858 

0.25 
0-3 
0-35 
0.4 

0-45 

0.52227  0.31424 
0.55151  0.37967 
0.58846  0.44616 
0.63452  0.51350 
0.69149  0.58116 

0.56223  0.29765 
0.59196  0.35856 

0.62928  0.41979 

0.67541  0.48093 
0.73188  0.54121 

0-59953  0.28085 
0,62935  0.33731 
0.66653  0.39344 
0.71212  0.44869 
0.76736  0.50211 

0.63419  0.26404 
0.66377  0.31618 
0.70039  0.36748 
0.74493  0.41714 

0.79836  0.46428 

0.66631  0.24741 
0.69534  0.29540 
0.73105  0.34205 
0.77413  0.38662 
0.82533  0.42806 

o-S 
0-55 
0.6 
0.65 
0.7 

0.76159  0.64805 
0.84752  0.71229 
0.95230  0.77067 
1.07907  0.81812 
1.23020  0.84688 

0.80050  0.59933 
0.88332  0.65320 
0.98247  0.69957 
1.09973  0.73362 
1.23587  0.74858 

0.83365  0.55229 
0.91249  0.59707 
1.00521  0.63346 
1.11262  0.65676 
1.23436  0.66157 

0.86173  0.50738 
0.93607  0.54434 
1.02195  0.57227 
1.11962  0.58738 
1.22793  0.58492 

0.88535  0.46492 
0.95480  0.49522 
1.03389  0.51635 

1.  12222  0.52508 
I.2I827  0.51757 

0-75 
0.8 
0.85 
0.9 
o-9S 

1.40579  0.84585 
1.60095  0.80073 
1.80225  0.69623 
1.98505  0.52197 
2.11599  0.28167 

1.38926  0.73549 
1.55398  0.68387 
1.71785  0.58390 

1.86183  0.43071 
1.96180  0.22977 

1.36782  0.64076 
1.50695  0.58681 
1.64135  0.49366 
1.75601  0.35949 
1.83417  0.19009 

1.34386  0.55951 
1.46173  0.50588 
1.57271  0.42040 
1.66532  0.30300 
1.72736  0.15910 

1.31896  0.48976 
1.41909  0.43803 
I.5II48  0.36034 
1.58713  0.25755 
I.637II  0.13449 

I.O 

1.05 
i.i 
i-iS 

1.2 

2.16395  °-o° 
2.11599  0.28167 
1.98505  0.52197 
1.80225  0.69623 
1.60095  0.80073 

1.99792  o.oo 
1.96180  0.22977 
1.86163  0.43071 

1.71785  0.58390 
1-55398  0.68387 

1.86202  o.oo 
1.83417  0.19009 

1.75601  0.35949 
1.64135  0.49366 
1.50695  0.58681 

1.74926  o.oo 
1.72736  0.15910 
1.66532  0.30300 
1.57271  0.42040 
1.46173  0.50588 

1.65462  o.oo 

1.63711  0.13449 
1.58713  0.25755 
1.51148  0.36034 
1.41909  0.43803 

I.2S 

i-3 
i-3S 
1.4 

i.  45 

1.40579  0.84585 
1.23020  0.84688 
1.07907  0.81812 
0.95230  0.77067 
0.84752  0.71229 

1.38926  0.73549 
1.23587  0.74858 
1.09973  0.73362 
0.98247  0.69957 
0.88332  0.65320 

1.36782  0.64076 

1.23436  0.66157 

1.11262  0.65676 
1.00521  0.63346 
0.91249  0.59707 

1.34386  0.55951 
1.22793  0.58492 
1.11962  0.58738 
1.02195  0.57227 
0.93607  0.54434 

1.31896  0.48976 
1.21827  0.51757 

1.  12222  0.52508 
1.03389  0.51635 
0.95480  O.49522 

i-5 
i-55 
1.6 
1.65 
i-7 

0.76159  0.64805 
0.69149  0.58116 
0.63452  0.51350 
0.58846  0.44616 
o-SS^i  0.37967 

0.80050  0.59933 
0.73188  0.54121 
0.67541  0.48093 
0.62928  0.41979 

0.59196  0.35856 

0.83365  0.55229 
0.76736  0.50211 
0.71212  0.44869 
0.66653  0.39344 
0.62935  0.33731 

0.86173  0.50738 
0.79836  0.46428 

0.74493  0.41714 
0.70039  0.36748 
0.66377  0.31618 

0.88535  0.46492 
0.82533  0.42806 
0.77413  0.38662 
0.73105  0.34205 
0.69534  0.29540 

1-75 
1.8 
1-85 
1.9 
i-9S 

0.52227  0.31424 
0.49964  0.24990 
0.48281  0.18651 
0.47119  0.12390 
0.46436  o.  06181 

0.56223  0.29765 
0.53910  0.23725 
0.52183  0.17737 

0.50987  0.11796 
0.50284  0.05889 

0-59953  0.28085 
0.57620  0.22437 
0.55872  0.16804 
0.54657  0.11189 
0.53941  0.05590 

0.63419  0.26404 
0.61094  0.21144 

0.59344  0.15863 
0.58125  0.10576 
0.57405  0.05288 

0.66631  O.2574I 
0.64336  0.19858 
O.626O2  O.I4925 
0.61390  0.09962 
0.60674  0.04984 

2.O 

0.46211  o.oo 

0.50052  o.oo 

0-53704  o.oo 

0.57167  o.oo 

0.60437  o.oo 

Note. 

Negative  quantities  are  in  heavy  type. 

Examples,  tanh  (0.6  +  i  0.6)  =  1.00521  +  i  0.63346. 

[76] 


TABLE  IX.     HYPERBOLIC  TANGENTS,     tanh  (x  +  iq)  =  u  +  iv.     CONTINUED 


x  =  0.75 


=  0.85 


*  =  0.9 


x  =  0.95 


o 

0.05 

O.I 

0.15 

0.2 

0.63515  o.oo 
0.63749  0.04684 
0.64456  0.09354 
0.65649  0.13997 
0.67352  0.18592 

0.66403  o.oo 
0.66633  0.04388 
0.67325  0.08758 
0.68490  0.13089 

0.70148  0.17357 

0.69107  o.oo 
0.69330  0.04099 
0.70002  0.08176 
0.71132  0.12206 
0.72736  0.16160 

0.71629  o.oo 

0.71845  0.03820 
0.72494  0.07614 

0.73582  0.11354 
0.75123  0.15008 

0.73978  o.oo 

0.74185  0.03551 
0.74807  0.07073 
0.75850  0.10537 
0.77321  0.13906 

0.25 
0-3 

o-3S 
0.4 

o-45 

0.69595  0.23112 
0.72420  0.27516 

0.75872  0.31749 
0.80005  0.35735 
0.84871  0.39368 

0.72325  0.21528 
0.75051  0.25559 
0.78364  0.29392 

0.82300  0.32949 
0.86893  0.36128 

0.74832  O.2OOOI 

0.77446  0.23683 

0.80603  0.27146 

0.84328  0.30314 

0.88638  0.33091 

0.77130  0.18537 

0.79620  0.21893 
0.82611  0.25018 
0.86117  0.27837 
0.90143  0.30261 

0.79231  0.17143 

0.81592  0.20198 
0.84411  0.23013 
0.87695  0.25520 
0.91438  0.27634 

0-5 
0-55 
0.6 
0.65 
0.7 

0.90515  0.42510 
0.96963  0.44978 
1.04203  0.46543 

1.  12161  0.46934 
1.20665  0.45847 

0.92167  0.38798 
0.98122  0.40796 
1.04722  0.41926 
1.11872  0.42057 
1.19395  0.40661 

0.93541  0.35357 

0.99018  0.36966 

1.05015  0.37751 
1.11427  0.37527 

1.  18081  0.36108 

0.94681  0.32181 
0.99700  0.33469 
1.05136  0.33985 
1.10885  0.33580 
1.16765  0.32108 

0.95624  0.29259 

I.O02II  0.30285 

1.05129  0.30593 

1.10271  0.30064 

1.15485  0.28588 

0-75 
0.8 
0.85 
0.9 
0-95 

1.29416  0.42977 

1.37961  0.38084 
1.45701  0.31065 
1.51945  0.22051 
1.56023  0.11463 

1.27012  0.37806 

1.34331  0.33238 

1.40862  0.26920 
1.46062  0.19000 
1.49428  0.09840 

1.24723  0.33335 

1.31017  0.29108 

1.36562  0.23434 

1.40931  0.16461 
1-43735  0.08499 

1.22572  0.29458 
1.28006  0.25573 
1.32742  0.20483 
1.36438  0.14330 
1.38796  0.07380 

1.20569  0.26087 

1.25279  0.22532 
1.29344  0.17968 
1.32493  0.12528 
1.34490  0.06438 

I.O 

1.05 
i.i 
i-iS 

1.2 

1.57443  o.oo 
1.56023  0.11463 
1.51945  0.22051 
1.45701  0.31065 
1.37961  0.38084 

1.50594  o.oo 
1.49428  0.09840 
1.46062  0.19000 
1.40862  0.26920 

1.34331  0.33238 

1.44703  o.oo 
1-43735  0.08499 
1.40931  0.16461 
1.36562  0.23434 
1.31017  0.29108 

1.39606  o.oo 
1.38796  0.07380 
1.36438  0.14330 
1.32742  0.20483 
1.28006  0.25573 

1.35175  o.oo 
1.34490  0.06438 
1.32493  0.12528 
1.29344  0.17968 
1.25279  0.22532 

1-25 

i-3 
i-3S 
1.4 

I-4S 

1.29416  0.42977 
1.20665  0.45847 
1.  12161  0.46934 
1.04203  0.46543 
0.96963  0.44978 

1.27012  0.37806 

1.19395  0.40661 

1.11872  0.42057 
1.04722  0.41926 
0.98122  0.40796 

1.24723  0.33335 
1.  18081  0.36108 
1.11427  0.37527 
1.05015  0.37751 
0.99018  0.36966 

1.22572  0.29458 
1.16765  0.32108 
1.10885  0.33580 
1.05136  0.33985 

0.99700  0.33469 

1.20569  0.26087 
1.15485  0.28588 
1.10271  0.30064 

1.05129  0.30593 

I.OO2II  0.30285 

i-5 
i-5S 
1.6 

1.65 
i-7 

0.90515  0.42510 

0.84871  0.39368 
0.80005  0.35735 
0.75872  0.31749 
0.72420  0.27516 

0.92167  0.38798 
0.86893  0.36128 
0.82300  0.32949 
0.78364  0.29392 

0.75051  0.25559 

0.93541  0.35357 
0.88638  0.33091 
0.84328  0.30314 
0.80603  0.27146 
0.77446  0.23683 

0.94681  0.32181 
0.90143  0.30261 
0.86117  0.27837 
0.82611  0.25018 
0.79620  0.21893 

0.95624  0.29259 
0.91438  0.27634 
0.87695  0.25520 
0.84411  0.23013 
0.81592  0.20198 

i-75 
1.8 

1.85 
1.9 
I-9S 

0.69595  0.23112 
0.67352  0.18592 
0.65649  0.13997 
0.64456  0.09354 
0.63749  0.04684 

0.72325  0.21528 
0.70148  0.17357 
0.68490  0.13089 
0.67325  0.08758 
0.66633  0.04388 

0.74832  O.200OI 

0.72736  0.16160 
0.71132  0.12206 
0.70002  0.08176 
0.69330  0.04099 

0.77130  0.18537 
0.75123  0.15008 
0.73582  0.11354 

0.72494  0.07614 
0.71845  0.03820 

0.79231  0.17143 
0.77321  0.13906 
0.75850  0.10537 

0.74807  0.07073 

0.74185  0.03551 

2.0 

0.63515  o.oo 

0.66403  o.oo 

0.69107  o.oo 

0.71629  o.oo 

0.73978  o.oo 

Note.  Negative  quantities  are  in  heavy  type. 

Examples,     tanh  (0.95  +  *  o)  =  0.73978  +  *o. 

tanh  (0.9  +  i  1.9)  =  0.72494  —  10.07614. 


[77] 


TABLE  IX.     HYPERBOLIC  TANGENTS,     tanh  (x  +  iq)  =  u  +  iv.     CONTINUED 


x  =  1.05 


x  =  1.15 


0 

0.05 

O.I 

0.15 

O.2 

0.76159  o.oo 
0-76357  0.03293 
0.76950  0.06556 
0.77943  0.09757 
0.79341  0.12858 

0.78181  o.oo 
0.78368  0.03048 
0.78932  0.06065 
0.79873  0.09016 
0.81195  0.11867 

0.80050  o.oo 
0.80227  0.02816 
0.80760  0.05599 
0.81648  0.08317 
0.82893  0.10932 

0.81775  o.oo 
0.81943  0.02597 

0.82444  0.05160 
0.83279  0.07658 
0.84447  0.10054 

0.83365  o.oo 
0.83522  0.02390 
0.83992  0.04748 
0.84775  0.07041 

0.85867  0.09233 

0.25 

o-3 
Q-3S 
0.4 

0-45 

0.81151  0.15821 
®-833  7  7  0.18598 
0.86022  0.21133 
0.89086  0.23361 
0.92554  0.25205 

0.82901  0.14575 
0.84991  0.17096 
0.87464  0.19377 
0.90311  0.21356 
o-935J5  0.22966 

0.84495  0.13405 
0.86450  0.15692 
0.88753  0.17742 
0.91392  0.19501 
0.94344  0.20906 

0.85945  0.12310 
0.87768  0.14383 

0.89907  0.16226 

0.92345  0.17789 

0.95058  0.19017 

0.87264  0.11288 
0.88958  0.13166 
0.90938  0.14823 
0.93186  0.16213 
0.95674  0.17287 

0-5 
o.SS 
0.6 
0.65 
0.7 

0.96403  0.26580 
1.00585  0.27392 
1.05030  0.27542 
1.09632  0.26933 
1.14516  0.25486 

0.97045  0.24130 
1.00852  0.24767 
1.04864  0.24798 
1.08984  0.24144 
1.13084  0.22747 

0.97574  0.21892 
1.01034  0.22389 
1.04654  0.22331 
1.08342  0.21658 
1.11984  0.20326 

0.98010  0.19852 
1.01151  0.20236 

1.04415  0.20115 

1.07718  0.19441 
1.10957  0.18182 

0.98368  0.17996 
1.01217  0.18288 
1.04160  0.18123 
1.07119  0.17461 
1.10003  0.16281 

0.75 
0.8 
0.85 
0.9 
0-95 

1.18715  0.23145 
1.22812  0.19904 
1.26319  0.15812 
1.29017  0.10993 
1.30721  0.05638 

1.17009  0.20572 
1.20585  0.17623 
1.23624  0.13955 
1.25949  0.09677 
1.27410  0.04956 

1.15445  0.18315 
1.18575  0.15637 
1.21219  0.12347 
1.23231  0.08544 
1.24492  0.04369 

1.14015  0.16330 
1.16763  0.13902 
1.19346  0.10950 
1.20821  0.07563 
1.21914  0.03863 

1.12709  0.14580 
1.15129  0.12380 

1.17152  0.09730 

1.  1  8680  0.06709 

1.19631  0.03424 

I.O 

i.  05 
i.i 
i-iS 

1.2 

1.31304  o.oo 
1.30721  0.05638 
1.29017  0.10093 
1.26319  0.15812 
1.22812  0.19904 

1.27908  o.oo 
1.27410  0.04956 

1.25949  0.09677 
1.23624  0.13955 
1.20585  0.17623 

1.24922  o.oo 
1.24492  0.04369 
1.23231  0.08544 
1.21219  0.12347 

1.18575  0.15637 

1.22286  o.oo 
1.21914  0.03863 
1.20821  0.07563 
1.19346  0.10950 
1.16763  0.13902 

1.19954  o.oo 
1.19631  0.03424 
1.  18680  0.06709 
1.17152  0.09730 
1.15129  0.12380 

1-25 

J-3 
1-35 
1.4 

i-45 

1.18715  0.23145 
1.14516  0.25486 
1.09632  0.26933 

1.05030  0.27542 
1.00585  0.27392 

1.17009  0.20572 
1.13084  0.22747 
1.08984  0.24144 
1.04864  0.24798 
1.00852  0.24767 

1.15445  0.18315 

1.11984  0.20326 
1.08342  0.21658 
1.04654  0.22331 
1.01034  0.22389 

1.14015  0.16330 
1.10957  0.18182 
1.07718  0.19441 
1.04415  0.20115 
1.01151  0.20236 

1.12709  0.14580 
1.10003  0.16281 
1.07119  0.17461 
1.04160  0.18123 
1.01217  0.18288 

i-5 
i-SS 
1.6 

1.65 
i-7 

0.96403  0.26580 
0.92554  0.25205 
0.89086  0.23361 
0.86022  0.21133 
0.83377  0.18598 

0.97045  0.24130 
0.93515  0.22966 
0.90311  0.21356 
0.87464  0.19377 

0.84991  0.17096 

0.97574  0.21892 
0.94344  0.20906 
0.91392  0.19501 
0.88753  0.17742 
0.86450  0.15692 

0.98010  0.19852 
0.95058  0.19017 

0.92345  0.17789 

0.89907  0.16226 
0.87768  0.14383 

0.98368  0.17996 
0.95674  0.17287 
0.93186  0.16213 
0.90938  0.14823 
0.88958  0.13166 

1-75 
1.8 
1.85 
1.9 
i-95 

0.81151  0.15821 
0.79341  0.12858 
0.77943  0.09757 
0.76950  0.06556 

0.76357  0.03293 

0.82901  0.14575 
0.81195  0.11867 
0-79873  0.09016 
0.78932  0.06065 
0.78368  0.03048 

0.84495  0.13405 
0.82893  0.10932 
0.81648  0.08317 
0.80760  0.05599 
0.80227  0.02816 

0.85945  0.12310 

0.84447  0.10054 
0.83279  0.07658 
0.82444  0.05160 
0.81943  0.02597 

0.87264  0.11288 
0.85867  0.09233 
0.84775  0.07041 
0.83992  0.04748 
0.83522  0.02390 

2JO 

0.76159  o.oo 

0.78181  o.oo 

0.80050  o.oo 

0.81775  o.oo 

0.83365  o.oo 

Note. 

Negative  quantities  are  in  heavy  type. 

Examples,  tanh  (1.2  +  to.  75)  =  1.12709  +  1*0.14580. 

[78] 


TABLE  IX.     HYPERBOLIC  TANGENTS,     tanh  (x  +  iq)  =  u+  iv.     CONTINUED 


*  =  1.25 


*  =  1.3 


*  =  i-35 


x  =  1.4 


*  =  1.45 


0 

0.84828  o.oo 

0.86172  o.oo 

0.87405  o.oo 

0.88535  o.oo 

0.89569  o.oo 

0.05 

0.84975  0.02197 

0.86309  0.02017 

0.87533  0.01849 

0.88653  0.01693 

0.89678  0.01549 

O.I 

0.85414  0.04363 

0.86719  0.04003 

0.87913  0.03668 

0.89006  0.03357 

0.90005  0.03070 

0.15 

0.86145  0.06464 

0.87398  0.05927 

0.88544  0.05428 

0.89591  0.04965 

0.90545  0.04537 

0.2 

0.87162  0.08468 

0.88344  0.07756 

0.89421  0.07097 

0.90401  0.06486 

0.91293  0.05923 

0.25 

0.88461  0.10339 

0.89548  0.09458 

0.90535  0.08644 

0.91429  0.07892 

0.92240  0.07199 

o-3 

0.90032  0.12039 

0.90791  0.10997 

0.91875  0.10036 

0.92663  0.09151 

0-93375  0.08338 

0-35 

0.91861  0.13528 

0.92686  0.12336 

0.93425  0.11240 

0.94087  0.10234 

0.94680  0.09312 

0.4 

0.93928  0.14765 

0.94585  0.13437 

0.95166  O.I222I 

0.95680  0.11108 

0.96137  0.10092 

0-45 

0.96207  0.15706 

0.96669  0.14262 

0.97069  O.I2945 

0.97417  0.11745 

0.97719  0.10654 

o-S 

0.98661  0.16307 

0.98903  0.14773 

O.99IOI  0.13381 

0.99263  0.12117 

0.99396  0.10971 

0-55 

1.01244  0.16528 

1.01243  0.14937 

I.OI2I9  0.13499 

1.01181  0.12199 

1.01133  0.11026 

0.6 

1.03897  0.16332 

1.03634  0.14722 

1-03375  0.13275 

1.03125  0.11972 

1.02885  0.10801 

0.65 

1.06550  0.15691 

1.06013  0.14109 

I.055IO  0.12693 

1.05042  0.11425 

1.04607  0.10288 

0.7 

1.09121  0.14591 

1.08308  0.13088 

1.07560  O.II749 

1.06875  0.10555 

1.06248  0.09488 

o-7S 

1.11521  0.13034 

1.10439  0.11665 

1.09457  0.10450 

1.08565  0.09371 

1.07756  0.08411 

0.8 

1.13656  0.11042 

1.12328  0.09862 

I.III3I  0.08820 

1.10052  0.07896 

1.09078  0.07077 

0.85 

1.15434  0.08662 

1.13895  0.07724 

I.I25I2  0.06897 

1.11277  0.06167 

1.  10166  0.05521 

0.9 

1.16772  0.05964 

1.15070  0.05311 

I.I355I  0.04738 

1.12192  0.04232 

1.10976  0.03785 

0-95 

1.17603  0.03041 

1.15799  0.02706 

I.I4I92  O.O24I2 

1.12758  0.02153 

1.11476  0.01925 

I.O 

1.17885  o.oo 

1.16047  o.oo 

1.14410  o.oo 

1.12950  o.oo 

1.11646  o.oo 

1.05 

1.17603  0.03041 

1.15799  0.02706 

1.14192  0.02412 

1.12758  0.02153 

1.11476  0.01925 

i.i 

1.16772  0.05964 

1.15070  0.05311 

1.13551  0.04738 

1.12192  0.04232 

1.10976  0.03783 

i-iS 

1.15434  0.08662 

1.13895  0.07724 

1.12512  0.06897 

1.11277  0.06167 

1.  10166  0.05521 

1.2 

1.13656  0.11042 

1.12328  0.09862 

1.11131  0.08820 

1.10052  0.07896 

1.09078  0.07077 

1-25 

1.11521  0.13034 

1.10439  0.11665 

1.09457  0.10450 

1.08565  0.09371 

1.07756  0.08411 

i-3 

1.09121  0.14591 

1.08308  0.13088 

1.07560  0.11749 

1.06875  0.10555 

1.06248  0.09488 

i-35 

1.06550  0.15691 

1.06013  0.14109 

1.05510  0.12693 

1.05042  0.11425 

1.04607  0.10288 

1.4 

1.03897  0.16332 

1.03634  0.14722 

1-03375  0.13275 

1.03125  0.11972 

1.02885  0.10801 

1-45 

1.01244  0.16528 

1.01243  0.14937 

1.01219  0.13499 

1.01181  0.12199 

1.01133  0.11026 

i-5 

0.98661  0.16307 

0.98903  0.14773 

0.99101  0.13381 

0.99263  0.12117 

0.99396  0.10971 

i-55 

0.96207  0.15706 

0.96669  0.14262 

0.97069  0.12945 

0.97417  0.17745 

0.97719  0.10654 

1.6 

0.93928  0.14765 

0.94585  0.13437 

0.95166  O.I222I 

0.95680  0.11108 

0.96137  0.10092 

1.65 

0.91861  0.13528 

0.92686  0.12336 

0.93425  O.II240 

0.94087  0.10234 

0.94680  0.09312 

i-7 

0.90032  0.12039 

0.90791  0.10997 

0.91875  O.IO036 

0.92663  0.09151 

0-93375  0.08338 

i-7S 

0.88461  0.10339 

0.89548  0.09458 

0.90535  0.08644 

0.91429  0.07892 

0.92240  0.07199 

1.8 

0.87162  0.08468 

0.88344  0.07756 

0.89421  O.O7O97 

0.90401  0.06486 

0.91293  0.05923 

1.85 

0.86145  0.06464 

0.87398  0.05927 

0.88544  O.O5428 

0.89591  0.04965 

0.90545  0.04537 

1.9 

0.85414  0.04363 

0.86719  0.04003 

0.87913  0.03668 

0.89006  0.03357 

0.90005  0.03070 

I-  95 

0.84975  0.02197 

0.86309  0.02017 

0.87533  0.01849 

0.88653  0.01693 

0.89678  0.01549 

2.0  0.84828     0.00 


0.86172   o.oo 


0.87405    o.oo 


0.88535 


0.89569   o.oo 


Note.  Negative  quantities  are  in  heavy  type. 

Examples,     tanh  (1.4  +  i  0.8)  =  1.10052  +  i  0.07896. 
tanh  (1.3  +  *  1.3)  =  1.08308  —  *  0.13088. 


[79] 


TABLE  IX.     HYPERBOLIC  TANGENTS,     tanh  (x  +  iq)  =  u  +  iv.     CONTINUED 


-  1.5 


=  i-55 


x  =  1.6 


=  1.65 


x  =  1.7 


0 

0.05 

O.I 

0.15 

O.2 

0.90515  o.oo 

0.90616  O.OI4I5 
0.90917  O.O28O4 
0.9I4I5  0.04143 
0.92104  0.05404 

0.91379  o.oo 
0.91471  0.01292 
0.91749  0.02560 
0.92208  0.03779 
0.92842  0.04927 

0.92167  o.oo 
0.92253  0.01178 
0.92508  0.02334 
0.92929  0.03445 

0.93511  0.04488 

0.92886  o.oo 
0.92964  0.01074 
0.93199  0.02127 
0.93586  0.03138 
0.94119  0.04086 

0.93541  o.oo 

0.93613  0.00979 
0.93828  0.01937 
0.94183  0.02857 
0.94671  0.03718 

O.25 

o-3 

o-35 
0.4 

o-45 

0.92975  0.06563 
0.94017  0.07593 
O.952I2  0.08468 
0.96542  0.09165 
0.97983  0.09660 

0.93641  0.05978 
0.94595  0.06909 
0.95689  0.07697 
0.96903  0.08320 
0.98214  0.08758 

0.94245  0.05442 

0.95118  0.06284 
0.96117  0.06993 

0.97223  0.07551 
0.98415  0.07938 

0.94791  0.04951 
0.95589  0.05712 
0.96501  0.06351 

0.97509  0.06850 

0.98592  0.07193 

0.95285  0.04502 

0.96015  0.05190 
0.96846  0.05766 
0.97763  0.06213 
0.98748  0.06517 

0-5 
0-55 
0.6 
0.65 
0.7 

0.99506  0.09933 
I.OIO76  0.09965 
I.O2657  0.09746 
I.O42O4  0.09268 
1.05675  0.08534 

0.99595  0.08092 
1.01016  0.09008 
1.02441  0.08796 
1.03834  0.08353 
1.05152  0.07680 

0.99668  0.08139 
1.00954  0.08142 
1.02240  0.07940 

1.03492  0.07530 

1.04676  0.06915 

0.99728  0.07367 

1.00891  0.07361 
1.02052  0.07169 
1.03179  0.06791 
1.04242  0.06230 

0-99777  0.06667 
1.00829  0.06655 
1.01877  0.06474 
1.02892  0.06126 
1.03847  0.05614 

0-75 
0.8 
0.85 
0.9 
0-95 

1.07022  0.07554 
I.O82OO  0.06349 
1.09167  0.04947 
1.09886  O.O339O 
I.IO329  O.OI723 

1.06357  0.06790 
1.07408  0.05700 
1.08269  0.04438 
1.08909  0.03038 
1.09302  0.01544 

1-05755  0.06107 

1.06693  0.05121 
1.07461  0.03984 
1.08030  0.02726 
1.08380  0.01385 

1.05209  0.05495 
1.06049  0.04604 

1.06734  0.03579 

1.07241  0.02448 

1.07554  0.01243 

1.04714  0.04948 
1.05466  0.04142 
1.06079  0.03218 

1.06533  O.02200 

1.06811  0.01117 

I.O 

1.05 
i.i 
i.i5 

1.2 

I.I0479  °'0° 
I.I0329  0.01723 
1.09886  O.O339O 
1.09167  O.O4947 
I.O82OO  O.OO349 

1.09436  o.oo 
1.09302  0.01544 
1.08909  0.03038 
1.08269  0.04438 
1.07408  0.05700 

1.08500  o.oo 
1.08380  0.01385 
1.08030  0.02726 
1.07461  0.03984 
1.06693  0.05121 

1.07659  o.oo 

1.07554  0.01243 

1.07241  0.02448 

1.06734  0.03579 

1.06049  0.04604 

1.06906  o.oo 
1.06811  0.01117 

1.06533  0.02200 

1.06079  0-03218 
1.05466  0.04142 

1.25 

i-3 
i-35 

1.4 

1-45 

I.O7O22  O.O75S4 
1.05675  0.08534 
I.O4204  0.09268 
I.O2657  0.09746 
I.OIO76  O.O9905 

1.06357  0.06790 
1.05152  0.07680 

1.03834  0.08353 

1.02441  0.08796 
1.01016  0.09008 

1.05755  0.06107 
1.04676  0.06915 
1.03492  0.07530 

1.02240  0.07940 
1.00954  0.08142 

1.05209  0.05495 

1.04242  0.06230 
1.03179  0.06791 
1.02052  0.07169 
1.00891  0.07361 

1.04714  0.04948 
1.03847  0.05614 
1.02892  0.06126 
1.01877  0.06474 
1.00829  0.06655 

i-5 
i-55 
1.6 

1.65 
i-7 

0.99506  0.09933 
0.97983  0.09660 
0.96542  O.O9I65 
O.952I2  0.08468 
0.94017  0.07593 

0.99595  0.08992 
0.98214  0.08758 
0.96903  0.08320 
0.95689  0.07697 
0.94595  0.06909 

0.99668  0.08139 
0.98415  0.07938 
0.97223  0.07551 
0.96117  0.06993 
0.95118  0.06284 

0.99728  0.07367 
0.98592  0.07193 
0.97509  0.06850 
0.96501  0.06351 

0.95589  0.05712 

0-99777  0.06667 
0.98748  0.06517 
0.97763  0.06213 
0.96846  0.05766 
0.96015  0.05190 

i-75 
1.8 
1.85 
1.9 

i-95 

0.92975  0.06563 
O.92IO4  O.O5404 
0.91415  0.04143 
0.90917  O.O28O4 
0.90616  O.OI4I5 

0.93641  0.05978 
0.92842  0.04927 
0.92208  0.03779 
0.91749  0.02560 
0.91471  0.01292 

0.94245  0.05442 

0.93511  0.04488 
0.92929  0.03445 
0.92508  0.02334 
0.92253  0.01178 

0.94791  0.04951 
0.94119  0.04086 
0.93586  0.03138 
0.93199  0.02127 
0.92964  0.01074 

0.95285  0.04502 
0.94671  0.03718 
0.94183  0.02857 
0.93828  0.01937 
0.93613  0.00979 

2.0 

0.90515  o.oo 

0.91379  o.oo 

0.92167  o.oo 

0.92886  o.oo 

0.93541  o.oo 

Note.  Negative  quantities  are  in  heavy  type. 

Examples,    tanh  (1.7  +  *Q-7)  =  1.03847  +  i  0.05614. 


tanh  (1.7  +  i  Q-7) 
tanh  (1.6  +  i  1.6) 


0.97223  —  »  0.07551. 


[80] 


TABLE  IX.     HYPERBOLIC  TANGENTS,     tanh  (x  +  iq)  =  u  +  iv.     CONTINUED 


x  =  1.75 


x  =  1.85 


x  =  1.9 


x  =  1.95 


o 
0.05 

O.I 

0.15 

0.2 

0.94138  o.oo 
0.94204  0.00891 
0.94400  0.01763 
0.94725  0.02600 

0.95172  0.03382 

0.94681  o.oo 
0.94741  0.00811 
0.94921  0.01604 
0.95218  0.02364 
0.95626  0.03074 

0.95175  o.oo 

0.95230  0.00737 
0.95394  0.01459 

0.95666  0.02149 
0.96038  0.02793 

0.95624  o.oo 
0.95674  0.00670 
0.95825  0.01326 
0.96072  0.01952 
0.96412  0.02537 

0.96032  o.oo 
0.96078  0.00609 
0.96215  0.01204 
0.96441  0.01773 
0.96750  0.02303 

0.2S 
0-3 

o-3S 
0.4 

0-45 

Q-95733  0.04092 
0.96399  0.04714 
0.97156  0.05233 
0.97991  0.05634 
0.98884  0.05904 

0.96139  O.037j8 
0.96746  0.04280 
0.97435  0.04748 
0.98194  0.05107 
0.99005  0.05348 

0.96506  0.03376 
0.97059  0.03885 

0.97686  0.04307 
0.98376  0.04629 
0.99111  0.04843 

0.96838  0.03065 

0.97342  0.03525 
0.97912  0.03905 
0.98538  0.04195 
0.99206  0.04386 

0.97138  0.02782 

0.97597  0.03198 

0.98116  0.03541 
0.98684  0.03801 
0.99289  0.03972 

o-5 
o-5S 
0.6 
0.65 
0.7 

0.99818  0.06034 
1.00769  0.06017 
1.01714  0.05848 
1.02629  0.05528 
1.03488  0.05061 

0.99851  0.05461 
1.00711  0.05440 
1.01565  0.05283 
1.02389  0.04989 
1.03162  0.04564 

0.99878  0.04942 
1.00656  0.04919 

1.01427  0.04773 

1.02170  0.04504 
1.02866  0.04118 

0.99900  0.04472 
1.00604  0.04448 
1.01300  0.04313 
1.01970  0.04067 
1.02596  0.03716 

0.99918  0.04047 
1.00555  0.04022 
1.01184  0.03897 
1.01788  0.03673 

1.02352  0.03354 

0-75 
0.8 
0.85 
0.9 
0-95 

1.04266  0.04457 
1.04940  0.03729 
1.05489  0.02895 
1.05895  0.01978 
1.06143  0.01004 

1.03861  0.04016 
1.04466  0.03358 
1.04958  0.02606 
1.05320  0.01780 
1.05543  0.00903 

1.03494  0.03621 
1.04037  0.03026 
1.04478  0.02347 
1.04804  0.01602 
1.05003  0.00813 

1.03162  0.03265 
1.03650  0.02727 
1.04046  0.02115 

1.04337  0.01443 

1.04516  0.00732 

1.02862  0.02946 
1.03300  0.02459 
1.03656  0.01906 
1.03918  0.01301 
1.04078  0.00659 

I.O 

1.05 
i.i 
1.13 

1.2 

1.06228  o.oo 
1.06143  0.01004 

1.05895  0.01978 
1.05489  0.02895 
1.04940  0.03729 

1.05619  o.oo 
1.05543  0.00903 

1.05320  0.01780 
1.04958  0.02606 
1.04466  0.03358 

1.05070  o.oo 
1.05003  0.00813 
1.04804  0.01602 
1.04478  0.02347 
1.04037  0.03026 

1.04576  o.oo 
1.04516  0.00732 

1.04337  0.01443 

1.04046  0.02115 
1.03650  0.02727 

1.04131  o.oo 
1.04078  0.00659 
1.03918  0.01301 
1.03656  0.01906 

1.03300  0.02459 

I.2S 

i-3 
i-35 
1.4 

1-45 

1.04266  0.04457 
1.03488  0.05061 
1.02629  0.05528 
1.01714  0.05848 
1.00769  0.06017 

1.03861  0.04016 
1.03162  0.04564 
1.02389  0.04989 
1.01565  0.05283 

1.00711  0.05440 

1.03494  0.03621 
1.02866  0.04118 
1.02170  0.04504 

1.01427  0.04773 

1.00656  0.04919 

1.03162  0.03265 
1.02596  0.03716 
1.01970  0.04067 
1.01300  0.04313 
1  .00604  0.04448 

1.02862  0.02946 

1.02352  0.03354 

1.01788  0.03673 
1.01184  o.o'3897 
1.00555  0.04022 

i-5 
i-5S 
1.6 

1.65 
i-7 

0.99818  0.06034 
0.98884  0.05904 
0.97991  0.05634 
0.97156  0.05233 
0.96399  0.04714 

0.99851  0.05461 
0.99005  0.05348 
0.98194  0.05107 
0.97435  0.04748 

0.96746  0.04280 

0.99878  0.04942 
0.99111  0.04843 
0.98376  0.04629 
0.97686  0.04307 
0.97059  0.03885 

0.99900  0.04472 
0.99206  0.04386 

0.98538  0.04195 

0.97912  0.03905 

0.97342  0.03525 

0.99918  0.04047 
0.99289  0.03972 
0.98684  0.03801 
0.98116  0.03541 
0-97597  0.03198 

i-7S 
1.8 

1.85 
1.9 

i-95 

0-95733  0.04092 
0.95172  0.03382 
0.94725  0.02600 
0.94400  0.01763 
0.94204  0.00891 

0.96139  0.03718 
0.95626  0.03074 
0.95218  0.02364 
0.94921  0.01604 
0.94741  0.00811 

0.96506  0.03376 
0.96038  0.02793 
0.95666  0.02149 

0.95394  0.01459 
0.95230  0.00737 

0.96838  0.03065 
0.96412  0.02537 
0.96072  0.01952 
0.95825  0.01326 
0.95674  0.00670 

0.97138  0.02782 
0.96750  0.02303 
0.96441  0.01773 
0.96215  0.01204 
0.96078  0.00609 

2.0        0.94138   o.oo  0.94681    o.oo  0.95175   o.oo 


0.95624   o.oo 


0.96032   o.oo 


Note. 


Negative  quantities  are  in  heavy  type. 


Examples,     tanh  (1.85  +  i  0.85)  =  1.04478  +  i  0.02347. 
tanh  (1.95  +  *  1.25)  =  1.02862  —  i  0.02946. 


[81] 


TABLE  IX.     HYPERBOLIC  TANGENTS,     tanh  (x  +  iq)  =  u  +  iv.     CONTINUED 


x  =  2.05 


x  =  2.15 


0 

0.05 

O.I 

0.15 

0.2 

0.96403  o.oo 

0.96445  0.00553 
0.96570  O.OIO94 

0-96775  0.01610 
0.97058  0.02090 

0.96740  o.oo 
0.96778  0.00502 
0.96892  0.00993 
0.97079  0.01461 
0.97336  0.01897 

0.97045  o.oo 
0.97080  0.00456 
0.97184  0.00901 

0.97354  0.01326 

0.97588  0.01721 

0.97323  o.oo 
0-97354  0.00413 
0.97449  0.00817 
0.97604  0.01203 
0.97816  0.01561 

0.97574  o.oo 

0.97603  0.00375 

0.97689  0.00741 
0.97830  0.01091 
0.98023  0.01415 

0.2S 

o-3 
°-35 
0.4 

o-4S 

0.97411  0.02524 
0.97827  0.02900 
0.98299  0.03209 
0.98815  0.03444 
0.99364  0.03596 

0.97657  0.02289 
0.98036  0.02630 
0.98464  0.02909 
0.98932  0.03120 
0.99430  0.03256 

0.97880  0.02076 
0.98224  0.02384 
0.98613  0.02636 
0.99037  0.02826 
0.99489  0.02948 

0.98081  0.01882 
0.98394  0.02161 
0.98747  0.02388 
0.99132  0.02559 
0.99541  0.02669 

0.98264  0.01706 
0.98548  0.01958 
0.98868  0.02163 
0.99217  0.02317 
0.99587  0.02416 

o-S 
o-SS 
0.6 
0.65 
0.7 

0.99933  0.03662 
1.00509  0.03638 
1.01077  0.03523 
1.01623  0.03318 
1.02131  0.03028 

0.99945  0.03314 
1.00466  0.03290 
1.00979  0.03184 
1.01471  0.02998 
1.01930  0.02734 

0.99955  0.02998 

1.00426  0.02976 
.00890  0.02878 
.01334  0.02709 
.01748  0.02469 

0.99963  0.02713 
1.00389  0.02691 
.00808  0.02602 

.OI2IO  0.02448 
.01583  0.02231 

0.99970  0.02455 
1.00355  0.02434 
1.00734  0.02353 

I.OI097  O.O22I2 
I.OI434  O.O2OI5 

0-75 
0.8 
0.85 
0.9 
Q-95 

1.02589  0.02658 
1.02984  0.02218 
1.03304  0.01719 
1-03539  0.01172 
1.03683  0.00594 

1.02343  0.02399 

1.02698  0.020OI 
1.02986  O.OI55O 
1.03197  0.01057 
1.03327  0.00536 

.02120  0.02166 
.02440  0.01806 
1.02699  0.01399 
1.02889  0.00954 
1.03005  0.00483 

.01919  O.OI956 
.022O7  O.OI63I 
.0244O  O.OI262 
.02611  O.OO86I 
.02715  0.00436 

1.01736  0.01767 
1.01996  O.OI472 
1.02206  O.OII40 
1.02360  O.O0777 
1.02454  0.00394 

I.O 

1.05 
i.i 
i-iS 

1.2 

1.03731  o.oo 
1.03683  0.00594 
1.03539  0.01172 
1.03304  0.01719 

1.02984  0.02218 

1.03370  o.oo 
1.03327  0.00536 
1.03197  0.01057 
1.02986  0.01550 

1.02698  O.O2OOI 

1.03045  o.oo 
1.03005  0.00483 
1.02889  0.00954 
1.02699  0.01399 
1.02440  0.01806 

1.02751  o.oo 
1.02715  0.00436 
1.02611  0.00861 
1.02440  0.01262 
1.02207  0.01631 

1.02486  o.oo 
1.02454  0.00394 
1.02360  0.00777 
1.02206  0.01140 
1.01996  0.01472 

1-25 
!-3 

i-3S 
1.4 

i.  45 

1.02589  0.02658 
1.02131  0.03028 
1.01623  0.03318 
1.01077  0.03523 
1.00509  0.03638 

1.02343  0.02399 
I.OI930  0.02734 
I.OI47I  0.02998 
I.O0979  0.03184 
I.OO466  0.0329O 

I.O2I2O  O.O2I66 
1.01748  0.02469 
I.OI334  0.02709 
1.00890  0.02878 
I.OO426  O.02976 

1.01919  0.01956 
1.01583  0.02231 

I.OI2IO  0.02448 
I.Oo8o8  0.02602 
1.00389  O.O269I 

1.01736  0.01767 
1.01434  0.02015 

I.OIO97  O.022I2 
1.00734  0.02353 
1.00355  0.02434 

i-5 
i-SS 
1.6 

1.65 
i-7 

0-99933  0.03662 
0.99364  0.03596 
0.98815  0.03444 
0.98299  0.03209 
0.97827  0.02900 

0.99945  0.03314 
0.99430  0.03256 
0.98932  O.O3I2O 
0.98464  O.02909 
0.98036  O.O263O 

0.99955  0.02998 
0.99489  0.02948 
0.99037  O.O2826 
0.98613  0.02636 
0.98224  0.02384 

0.99963  0.02713 
0.99541  0.02669 
0.99132  0.02559 
0.98747  O.02388 

0.98394  0.02161 

0.99970  0.02455 
0.99587  O.O24I6 
0.99217  O.O23I7 
0.98868  O.02I63 
0.98548  O.OI958 

1-75 
1.8 
1.85 
1.9 
i  -95 

0.97411  0.02524 
0.97058  0.02090 
0.96775  0.01610 
0.96570  0.01094 
0.96445  0.00553 

0-97657  0.02289 
0.97336  0.01897 
0.97079  O.OI46I 
0.96892  O.OO993 
0.96778  O.OO5O2 

0.97880  O.O2O76 
0.97588  O.OI72I 
0.97354  0.01326 

0.97184  jo.  0090  1 
0.97080  0.00456 

0.98081  0.01882 
0.97816  0.01561 
0.97604  0.01203 
0.97449  0.00817 

0.97354  0.00413 

0.98264  O.OI7O6 
0.98023  O.OI4I5 
0.97830  O.OIO9I 
0.97689  O.0074I 
0.97603  0.00375 

2.0 

0.96403  o.oo 

0.96740  o.oo 

0.97045  o.oo 

0.97323  o.oo 

0.97574  o.oo 

Note.  Negative  quantities  are  in  heavy  type. 

Examples,     tanh  (2.2  +  *  o)  =  o  97574  +  i  o. 

tanh  (2.15  -f  i  1.15)  =  1.02440  —  £0.01262. 


[82] 


TABLE  IX.     HYPERBOLIC  TANGENTS,     tanh  (x  +  iq)  =  u  +  iv.     CONTINUED 


x  =  2.25 


x  =  2.3 


=  2.35 


2.4 


*  =  2.45 


o 
0.05 

O.I 

0.15 

O.2 

0.97803  o.oo 
0.97829  0.00340 
0.97907  0.00672 
0.98035  0.00989 
0.98210  0.01283 

0.98010  o.oo 
0.98034  0.00308 
0.98105  0.00610 
0.98221  0.00897 
0.98380  0.01163 

0.98197  o.oo     0.98367  o.oo 
0.98219  0.00280   0.98387  0.00253 
0.98283  0.00553   0.98446  0.00501 
0.98389  0.00813   0.98541  0.00736 
0.98534  0.01054   0.98673  0.00955 

0.98522  o.oo 
0.98540  0.00230 
0.98592  0.00454 

0.98680  0.00667 
0.98799  0.00865 

0.25 

°-3 
°-35 

0.4 

0-45 

0.98429  0.01547 
0.98687  0.01774 
0.98977  0.01960 
0.99294  0.02098 
0.99629  0.02187 

0.98579  0.01401 
0.98813  0.01607 
0.99076  0.01775 
0.99363  0.01900 
0.99667  0.01979 

0.98714  0.01270   0.98836  0.01150 
0.98926  0.01456   0.99029  0.01319 
0.99165  0.01607   0.99245  0.01456 
0.99425  0.01720   0.99481  0.01557 
0.99700  0.01791   0.99730  0.01621 

0.98947  0.01042 
0.99121  0.01194 
0.99317  0.01318 
0.99531  0.01410 
O-99757  0.01468 

o-5 
0-55 
0.6 
0.65 
0.7 

O.99975  O.O2222 
1.00324  O.O22O2 
I.OO666  O.O2I27 
I.OO994  O.O2OOO 
I.OI298  O.OI82I 

0.99980  O.O2OIO 
1.00295  0.01992 
I.O06O4  O.OI924 
1.00900  O.Ol8o8 
I.OII75  0.01646 

0.99984  0.01819   0.99986  0.01646 
1.00269  0.01802   1.00245  0.01630 
.00549  0.01740   1.00498  0.01573 
.00816  0.01634   1.00739  0.01478 
.01064  0.01487   1.00963  0.01345 

0.99989  0.01489 
1.00222  0.01474 
1.00451  0.01423 
1.00693  0.01336 
1.00872  0.01216 

0-75 
0.8 
0.85 
0.9 
0-95 

I.OI57I  0.01596 
I.OI8O5  O.OI33O 
I.OI994  O.OIO29 
I.02I33  O.OO7OI 
I.022I8  0.00355 

I.OI42I  O.OI442 
I.OI632  O.OI2OI 
I.OI8O3  0.00929 
I.OI928  0.00633 
I.O2006  O.OO322 

.01286  0.01303   1.01164  0.01178 
.01477  0.01085   1.01336  0.00981 
1.01631  0.00839   1-OI475  0.00758 
1.01744  0.00572   1.01576  0.00517 
1.01812  0.00290   1.01639  0.00262 

1.01053  0.01064 
1.01208  0.00886 
1.01333  0.00685 
1.01425  0.00467 
1.01482  0.00237 

.0 

•°5 
.1 

•IS 

.2 

1.02247  o.oo 
1.02218  0.00355 
1.02133  0.00701 
1.01994  0.01029 
1.01805  0.01330 

1.02031  o.oo 

I.O20O6  O.OO322 
I.OI928  O.OO633 
I.OI8O3  O.OO929 
I.OI632  O.OI2OI 

1.01836  o.oo     1.01659  o.oo 
1.01812  0.00290   1.01639  0.00262 
1.01744  0.00572   1.01576  0.00517 
1.01631  0.00839   1.01475  0.00758 
1.01477  0.01085   1.01336  0.00981 

1.01500  o.oo 
1.01482  0.00237 
1.01425  0.00467 

1.01333  0.00685 

1.  01208  0.00886 

•25 

•3 
•35 
•4 
J-45 

1.01571  0.01596 
1.01298  0.01821 

I.OO994  O.O2OOO 
I.00666  0.02127 
1.00324  O.O22O2 

I.OI42I  O.OI442 
I.OII75  0.01646 
I.OO9OO  O.OlSoS 
1.00604  O.OI924 
I.OO295  O.OI992 

1.01286  0.01303   1.01164  0.01178 
1.01064  0.01487   1.00963  0.01345 
1.00816  0.01634   1.00739  0.01478 
1.00549  0.01740   1.00498  0.01573 
1.00269  0.01802   1.00245  0.01630 

1.01053  0.01064 
1.00872  0.01216 
1.00693  0.01336 
1.00451  0.01423 

1.00222  O.OI474 

i-5 
i-55 
1.6 

1.65 
i-7 

0-99975  0.02222 
0.99629  O.02I87 
0.99294  O.O2098 
0.98977  O.OI96O 
0.98687  O.OI774 

0.99980  O.O20IO 
0.99667  O.OI979 
0.99363  O.OI9OO 
0.99076  O.OI775 
0.98813  O.OI6O7 

0.99984  0.01819   0.99986  0.01646 
0.99700  0.01791   0.99730  0.01621 
0.99425  0.01720   0.99481  0.01557 
0.99165  0.01607   0.99245  0.01456 
0.98926  0.01456   0.99029  0.01319 

0.99989  0.01489 
0.99757  0.01468 
0.99531  O.OI4IO 
0.99317  O.OI3I8 
O.99I2I  O.OII94 

1-75 
1.8 
1.85 
1.9 
1-95 

0.98429  O.OI547 
0.98210  O.OI283 
0.98035  0.00989 
0.97907  O.OO672 
0.97829  O.OO340 

0.98579  O.OI4OI 
0.98380  O.OII63 
0.98221  O.O0897 
0.98105  O.OO6IO 
0.98034  O.OO3O8 

0.98714  0.01270   0.98836  0.01150 
0.98534  0.01054   0.98673  0.00955 
0.98389  0.00813   0.98541  0.00736 
0.98283  0.00553   0.98446  0.00501 
0.98219  0.00280   0.98387  0.00253 

0.98947  O.OI042 
0.98799  0.00865 
0.98680  0.00667 
0.98592  O.OO454 
0.98540  O.OO23O 

2.0 

0.97803  o.oo 

0.98010  o.oo 

0.98197  o.oo     0.98367  o.oo 

0.98522  o.oo 

Note. 

Negative  quantities  are  in  heavy  type. 

Examples,  tanh  (2.25  +  £0.25)  =  0.98429  +  i  0.01547. 

tanh  (2.45  +  *  1.45)  =  1.00222  —  *  0.01474. 


[83] 


TABLE  IX.     HYPERBOLIC  TANGENTS,     tanh  (x  +  iq)  =  u  +  iv.     CONTINUED 


=  2.5 


*  =  2.55 


*  =   2.6 


x  =  2.65 


x  =  2.7 


0 

0.05 

O.I 

0.15 

O.2 

o.  8661  o.oo 

0.98678  O.O0208 
0.98726  O.004II 
0.98805  O.006O5 
0.98913  0.00784 

0.98788  o.oo 
0.98803  0.00189 
0.98846  0.00373 
0.98918  0.00548 
0.99016  0.00710 

0.98903  o.oo 

0.98916  O.OOI7I 
0.98956  0.00337 
O.9902I  0.00496 
0.99109  0.00643 

0.99007  o.oo 
0.99019  0.00155 
0.99055  0.00306 
0.99113  0.00449 
0.99194  0.00582 

0.99101  o.oo 
0.99112  0.00140 
0.99144  0.00277 
0.99198  0.00407 
0.99270  0.00527 

0.25 

o-3 
o-3S 
0.4 

Q-4S 

0.99047  O.00944 
0.99205  0.01082 
0.99383  O.OII93 
0.99576  O.OI276 
0.99781  0.01328 

0.99138  0.00855 
0.99281  0.00979 
0.99442  0.01080 
0.99617  0.01155 

0.99802  O.OI2O2 

O.9922O  O.OO774 
0.99350  0.00887 
0.99495  0.00978 
0.99654  O.OI046 
0.99822  O.OI088 

0.99294  0.00701 
0.99412  0.00803 
0.99544  0.00886 
0.99688  0.00947 
0.99839  0.00985 

0.99361  0.00635 

0.99468  0.00727 
0.99588  0.00802 
0.99718  0.00857 
0.99855  0.00891 

o-5 
o-SS 
0.6 
0.65 
0.7 

0.99991  0.01348 
1.00202  O.OI334 
I.004O9  O.OI287 
I.Oo6o6  O.OI2O8 
1.00789  O.OIO99 

0.99993  O.OI2I9 
I.OOI84  O.OI207 
I.OO37I  O.OII64 
1.00549  0.01093 
I.OO7I4  0.00994 

0.99994  O.OIIO3 
I.OOI67  O.OIO92 
1.00336  0.01053 
1.00498  0.00988 
1.00647  0.00898 

0.99995  0.00998 
1.00151  0.00988 
1.00304  0.00952 
1.00450  0.00894 
1.00585  0.00812 

0.99996  0.00903 
1.00137  0.00893 
1.00276  0.00862 
1.00408  0.00808 

1.00530  0.00735 

0-75 
0.8 
0.85 
0.9 
o-9S 

1.00953  0.00962 
I.OIO93  O.00§01 
I.  OI2O6  0.00619 
I.OI289  0.00422 
I.OI34O  O.O02I4 

I.O0862  O.OO87O 
1.00989  O.OO726 
I.OIOgi  0.00560 

1.01166  0.00381 
i.oi2ii  0.00193 

I.O078o  O.00786 
1.00895  0.00654 
I.OOgS?  0.00506 
I.OI054  0.00345 
1.01096  O.OOI75 

1.00706  0.00711 
1.00809  0.00592 
1.00893  0-°°457 
1.00954  0.00312 
1.00991  0.00158 

1.00639  0.00643 
1.00732  0.00535 

1.00807  0.00413 
1.00862  0.00282 
1.60896  0.00143 

I.O 

1.05 
i.i 
i-iS 

1.2 

1.01357  o.oo 

I.OI34O  O.OO2I4 
1.01289  O.00422 

1.01206  0.00619 
1.01093  0.00801 

1.01227  °-°° 

1.01211   O.OOI93 
I.OII66   O.0038I 
I.OIOgi   O.OO56O 
1.00989   O.OO726 

I.  OHIO  O.OO 

1.01096  0.00175 
1.01054  0.00345 
1.00987  0.00506 
1.00895  0.00654 

1.01003  o.oo 
1.00991  0.00158 
1.00954  0.00312 
1.00893  0.00457 
1.00809  0.00592 

1.00907  o.oo 
1.00896  0.00143 
1.00862  0.00282 
1.00807  0.00413 
1.00732  0.00535 

1-25 

i-3 
i-3S 

1.4 

145 

1.00953  0.00962 
1.00789  0.01099 
1.00606  0.01208 
1.00409  0.01287 

1.00202  O.OI334 

I.OO862   O.0087O 
I.OO7I4  O.OO994 
I.OO549  O.OIO93 
I.O037I   O.OII64 
I.OOl84   O.O12O7 

1.00780  0.00786 
1.00647  0.00898 
1.00498  0.00988 
1.00336  0.01053 
1.00167  0.01092 

1.00706  0.00711 
1.00585  0.00812 
1.00450  0.00894 
1.00304  0.00952 
1.00151  0.00988 

1.00639  0.00643 

1.00530  0.00735 

1.00408  0.00808 
1.00276  0.00862 
1.00137  0.00893 

i-5 
i-55 
1.6 
1.65 
i-7 

0.99991  O.OI348 
0.99781  O.OI328 
0.99576  O.OI276 
0-99383  O.OII93 
0.99205  O.OI082 

O-99993   O.OI2I9 
0.99802  O.OI2O2 
0.99617  O.OII55 
0.99442  O.OI08O 
0.99281  O.OO979 

0.99994  0.01103 
0.99822  0.01088 
0.99654  0.01046 
0.99495  0.00978 
0.99350  0.00887 

0.99995  0.00998 
0.99839  0.00985 
0.99688  0.00947 
0.99544  0.00886 
0.99412  0.00803 

0.99996  0.00903 
0.99855  0.00891 
0.99718  0.00857 
0.99588  0.00802 
0.99468  0.00727 

i-7S 
1.8 
1.85 
1.9 
i-9S 

0.99047  0.00944 
0.98913  0.00784 
0.98805  O.00605 
0.98726  O.OO4II 
0.98678  O.00208 

0.99138  0.00855 
0.99016  O.OO7IO 
0.98918  O.OO548 
0.98846  O.00373 
0.98803  O.OOI89 

0.99220  0.00774 
0.99109  0.00643 
0.99021  0.00496 
0.98956  0.00337 
0.98916  0.00171 

0.99294  0.00701 
0.99194  0.00582 
0.99113  0.00449 
0.99055  0.00306 
0.99019  0.00155 

0.99361  0.00635 
0.99270  0.00527 
0.99198  0.00407 
0.99144  0.00277 
0.99112  0.00140 

2.O 

0.98661  o.oo 

0.98788  o.oo 

0.98903  o.oo 

0.99007  o.oo 

0.99101  o.oo 

Note.  Negative  quantities  are  in  heavy  type. 

Examples,     tanh  (2.60  +  i  0.35)  =  0.99495  +  *  0.00978. 
tanh  (2.70  +  *  1.35)  =  1.00408  —  i  0.00808. 


[84] 


TABLE  IX.     HYPERBOLIC  TANGENTS,     tanh  (x  +  iq)  =  u  +  iv.     CONTINUED 


x  =  2.75 


x  =  2.80 


x  =  2.85 


=  2.90 


x  =  2.95 


0 

0.99186  o.oo 

0.99263  o.oo 

0-99333  o.oo 

0.99396  o.oo 

0.99454  o.oo 

0.05 

0.99196  0.00127 

0.99272  0.00115 

0.99341  0.00104 

0.99404  0.00094 

0.99460  0.00085 

O.I 

0.99225  0.00251 

0.99299  0.00227 

0.99365  0.00206 

0.99426  0.00186 

0.99480  0.00168 

0.15 

0.99274  0.00368 

0.99343  0.00334 

0.99405  0.00302 

0.99462  0.00273 

0.99513  0.00248 

0.2 

0.99340  0.00477 

0.99403  0.00432 

0.99459  0.00391 

0.99511  0.00354 

0-99557  0.00321 

0.25 

0.99422  0.00575 

0.99477  0.00519 

0.99527  0.00471 

0.99572  0.00426 

0.99613  0.00386 

0-3 

0.99519  0.00658 

0.99564  0.00596 

0.99606  0.00539 

0.99644  0.00488 

0.99678  0.00442 

o-3S 

0.99627  0.00726 

0.99663  0.00657 

0.99695  0.00594 

0.99724  0.00538 

0.99750  0.00487 

0.4 

0.99745  0.00775 

0.99769  0.00702 

0.99791  0.00635 

0.99811  0.00575 

0.99830  0.00520 

0-45 

0.99869  0.00806 

0.99882  0.00730 

0.99893  0.00660 

0.99904  0.00598 

0.99913  0.00541 

o-5 

0.99997  0.00817 

0.99997  0.00740 

0.99998  0.00669 

0.99998  0.00606 

0.99999  0.00548 

o-55 

1.00125  0.00808 

1.00113  0.00731 

1.00103  0.00662 

1.00093  0.00599 

1.00084  0.00542 

0.6 

1.00250  0.00779 

1.00226  0.00705 

1.00205  0.00638 

1.00186  0.00577 

1.00168  0.00522 

0.65 

1.00369  0.00731 

1.00334  0.00661 

1.00303  0.00598 

1.00274  0.00541 

1.00248  0.00489 

0.7 

1.00479  0.00664 

1.00434  0.00601 

1.00393  0.00544 

I-0035S  0.00480 

1.00322  0.00445 

o.7S 

1.00578  0.00581 

1.00523  0.00525 

1.00473  0.00475 

1.00428  0.00430 

1.00387  0.00389 

0.8 

1.00662  0.00484 

1.00599  0-00437 

1.00542  0.00396 

1.00490  0.00358 

1.00444  0.00324 

0.85 

.00730  0.00374 

1.  0066  1  0.00338 

1.00598  0.00306 

1.00540  0.00276 

1.00489  0.00250 

0.9 

.00780  0.00255 

1.00706  0.00230 

1.00638  0.00208 

1.00577  0.00188 

1.00522  0.00170 

o-95 

.00810  0.00129 

1.00733  0.00117 

1.00663  0.00105 

1.00600  0.00095 

1.00542  0.00086 

I.O 

.00821  o.oo 

1.00742  o.oo 

1.00671  o.oo 

1.00607  o.oo 

1.00549  o.oo 

'1.05 

.00810  0.00129 

1.00733  0.00117 

1.00663  0.00105 

1.00600  0.00095 

1.00542  0.00086 

i.i 

.00780  0.00255 

1.00706  0.00230 

1.00638  0.00208 

1.00577  0.00188 

1.00522  0.00170 

i-iS 

.00730  0.00374 

1.  00661  0.00338 

1.00598  0.00306 

1.00540  0.00276 

1.00489  0.00250 

1.2 

1.00662  0.00484 

1.00599  0.00437 

1.00542  0.00396 

1.00490  0.00358 

1.00444  0.00324 

I-2S 

1.00578  0.00581 

1.00523  0.00525 

1.00473  0.00475 

1.00428  0.00430 

1.00387  0.00389 

i-3 

1.00479  0.00664 

1.00434  0.00601 

1.00393  0.00544 

1  -003  5  5  0.00480 

1.00322  0.00445 

i-35 

1.00369  0.00731 

1.00334  0.00661 

1.00303  0.00598 

1.00274  0.00541 

1.00248  0.00489 

1.4 

1.00250  0.00779 

1.00226  0.00705 

1.00205  0.00638 

1.00186  0.00577 

i  .00168  0.00522 

i  -45 

1.00125  0.00808 

1.00113  0.00731 

1.00103  0.00662 

1.00093  0.00599 

1.00084  0.00542 

i-5 

0.99997  0.00817 

0.99997  0.00740 

0.99998  0.00669 

0.99998  0.00606 

0.99999  0.00548 

i-55 

0.99869  0.00806 

0.99882  0.00730 

0.99893  0.00660 

0.99904  0.00598 

0.99913  0.00541 

1.6 

0.99745  0.00775 

0.99769  0.00702 

0.99791  0.00635 

0.99811  0.00575 

0.99830  0.00520 

1.65 

0.99627  0.00726 

0.99663  0.00657 

0.99695  0.00594 

0.99724  0.00538 

0.99750  0.00487 

i-7 

0.99519  0.00658 

0.99564  0.00596 

0.99606  0.00539 

0.99644  0.00488 

0.99678  0.00442 

1-75 

0.99422  0.00575 

0.99477  0.00519 

0.99527  0.00471 

0.99572  0.00426 

0.99613  0.00386 

1.8 

0.99340  0.00477 

0.99403  0.00432 

0.99459  0.00391 

0.99511  0.00354 

0-99557  0.00321 

1.85 

0.99274  0.00368 

0-99343  0.00334 

0.99405  0.00302 

0.99462  0.00273 

0.99513  0.00248 

1.9 

0.99225  0.00251 

0.99299  0.00227 

0.99365  0.00206 

0.99426  0.00186 

0.99480  0.00168 

1-95 

0.99196  0.00127 

0.99272  0.00115 

0.99341  0.00104 

0.99404  0.00094 

0.99460  0.00085 

2.O  0.99186     O.OO 


0.99263   o.oo 


0-99333    o.oo 


0.99396   o.oo 


0.09454   o.oo 


Note. 


Negative  quantities  are  in  heavy  type. 


Examples,     tanh  (2.9  +  i  0.9)  =  1.00577  +  *  0.00188. 

tanh  (2.95  +  i  1.95)  =  0.99460  —  *  0.00085. 


[85] 


TABLE  IX.     HYPERBOLIC  TANGENTS,     tanh  (x  +  iq)  =  u  +  iv.     CONTINUED 


=  3-0 


*  =  3-°5 


x  =  3.10 


x  =  3.20 


0 

0.05 

O.I 

0.15 

O.2 

0.99505  o.oo 

0.99512  O.OOO77 
0.99530  0.00153 
0.99559  0.00224 
0.99599  O.O029O 

0.99552  o.oo 
0.99558  0.00070 
0.99574  0.00138 
0.99601  0.00203 
0.99637  0.00263 

0.99595  o.oo 
0.99600  0.00063 
0.99615  0.00125 
0.99639  0.00184 
0.99672  0.00238 

0.99633  o.oo 
0.99638  0.00057 
0.99651  0.00113 
0.99673  0.00166 
0.99703  0.00215 

0.99668  o.oo 
0.99672  0.00052 

0.99684  O.OOIO2 

0.99704  0.00150 
0.99731  0.00195 

0.25 
0-3 
0-35 
0.4 

0-45 

0.99649  0.00349 
0.99708  O.O040O 
0.99774  O.OO44I 
0.99846  O.0047I 
0.99921  0.00489 

0.99683  0.00316 
0.99736  0.00362 
0.99796  0.00399 
0.99861  0.00426 
0.99929  0.00443 

0.99713  0.00286 
0.99761  0.00328 
0.99815  0.00361 
0.99874  0.00386 
0.99936  0.00401 

0.99740  0.00259 
0.99784  0.00297 
0.99833  0.00327 
0.99886  0.00349 
0.99942  0.00363 

0.99765  0.00234 

0.99805  0.00268 
0.99849  0.00296 
0.99897  0.00316 
0.99947  0.00328 

0-5 
0-55 

0.6 

0.65 
0.7 

0.99999  0.00496 
I.OOO76  0.00490 
I.OOI52  0.00472 
1.00224  0.00443 
I.OO29I  O.OO4O2 

0.99999  0.00449 
1.00069  0.00443 
1.00138  0.00427 
1.00203  0.00401 
1.00263  0.00364 

0.99999  0.00406 
1.00063  0.00401 
1.00125  0.00387 
1.00184  0.00362 
1.00238  0.00329 

0.99999  0.00367 
1.00057  0.00363 
1.00113  0.00350 
1.00166  0.00328 
1.00216  0.00298 

0.99999  0.00332 

1.00051  0.00328 

I.OOI02  O.OO3I6 

1.00151  0.00297 
1.00195  0.00269 

0-75 
0.8 
0.85 
0.9 
0-95 

I.0035I  0.00352 
I.OO4OI  O.OO293 
1.00443  O.00226 
1.00473  0.00154 
1.00491  0.00078 

1.00317  0.00318 
1.00363  0.00265 
1.00400  0.00205 
1.00426  0.00139 
1.00444  0.00071 

1.00287  0.00288 
1.00329  0.00239 
1.00362  0.00185 
1.00387  0.00126 
1.00402  0.00064 

1.00260  0.00260 
1.00297  0.00217 
1.00328  0.00167 
1.00350  0.00114 
1.00363  0.00058 

1.00235  0.00236 
1.00269  0.00196 
1.00297  0.00151 
1.00317  0.00103 
1.00329  0.00052 

.0 

•OS 
.1 

•IS 

.2 

1.00497  o.oo 
1.00491  0.00078 
1.00473  0.00154 
1.00443  0.00226 
1.00401  0.00293 

1.00450  o.oo 
1.00444  0.00071 
1.00426  0.00139 
1.00400  0.00205 
1.00363  0.00265 

1.00407  o.oo 
1.00402  0.00064 
1.00387  0.00126 
1.00362  0.00185 
1.00329  0.00239 

1.00368  o.oo 
1.00363  0.00058 
1.00350  0.00114 
1.00328  0.00167 
1.00297  0.00217 

1.00333  o.oo 
1.00329  0.00052 
1.00317  0.00103 
1.00297  0.00151 
1.00269  0.00196 

•25 

•3 

•35 
1.4 

i-45 

1.00351  0.00352 

1.00291  0.00402 
1.00224  0.00443 
1.00152  0.00472 
1.00076  0.00490 

1.00317  0.00318 
1.00263  0.00364 
1.00203  0.00401 
1.00138  0.00427 
1.00069  0.00443 

1.00287  0.00288 
1.00238  0.00329 
1.00184  0.00362 
1.00125  0.00387 
1.00063  0.00401 

1.00260  0.00260 
1.  00216  0.00298 
1.00166  0.00328 
1.00113  0.00350 
1.00057  0.00363 

1.00235  0.00236 
1.00195  0.00269 
1.00151  0.00297 

I.OOI02  O.O03I6 

1.00051  0.00328 

i-5 
i-SS 
1.6 

1.65 
i-7 

0.99999  0.00496 
0.99921  0.00489 
0.99846  0.00471 
0.99774  0.00441 
0.99708  0.00400 

0.99999  0.00449 
0.99929  0.00443 
0.99861  0.00426 
0.99796  0.00399 
0.99736  0.00362 

0.99999  0.00406 
0.99936  0.00401 
0.99874  0.00386 
0.99815  0.00361 
0.99761  0.00328 

0.99999  0.00367 
0.99942  0.00363 
0.99886  0.00349 
0.99833  0.00327 
0.99784  0.00297 

0.99999  0.00332 
0.99947  0.00328 
0.99897  0.00316 
0.99849  0.00296 
0.99805  0.00268 

i-75 
1.8 
1.85 
1.9 

i-95 

0.99649  0.00349 
0.99599  0.00290 
0-99559  0.00224 
0.99530  0.00153 
0.99512  0.00077 

0.99683  0.00316 
0.99637  0.00263 
0.99601  0.00203 
0.99574  0.00138 
0.99558  0.00070 

0.99713  0.00286 
0.99672  0.00238 
0.99639  0.00184 
0.99615  0.00125 
0.99600  0.00063 

0.99740  0.00259 
0.99703  0.00215 
0.99673  0.00166 
0.99651  0.00113 
0.99638  0.00057 

0.99765  0.00234 
0.99731  0.00195 
0.99704  0.00150 

0.99684  O.OOI02 

0.99672  0.00052 

2.O 

0-99505  o.oo 

0.99552  o.oo 

0.99595  o-o° 

o-99633  o-00 

0.99668  o.oo 

Note.  Negative  quantities  are  in  heavy  type. 


Examples. 


tanh  (3.0  +  *  i.oo) 
tanh  (3.0  +  *  1.50) 


1.00497  +  io. 
0.99999  ~~  »  0.00496. 


[86] 


TABLE  IX.     HYPERBOLIC  TANGENTS,     tanh  (x  +  iq)  =u+  iv.     CONTINUED 


=  3-25 


*  =  3-30 


*  =  3-35 


x  =  3.40 


=  3-45 


o 

0.99700  o.oo 

0.99728  o.oo 

0.99754  o.oo 

0.99777  o.oo 

0.99799  o.oo 

0.05 

0.99704  0.00047 

0.99732  0.00042 

0.99757  0.00038 

0.99780  O.OOO35 

0.99801  0.00031 

O.I 

0.99715  0.00093 

0.99742  0.00084 

0.99766  0.00076 

0.99788  0.00069 

0.99809  0.00062 

0.15 

0.99732  0.00136 

0.99758  0.00123 

0.99781  O.OOII2 

0.99802  o.ooioi 

0.99821  0.00091 

0.2 

0.99757  0.00176 

0.99780  0.00160 

0.99801  0.00144 

0.99820  0.00131 

0.99837  0.00118 

0.25 

0.99787  O.O02I2 

0.99808  0.00192 

0.99826  O.OOI74 

0.99843  0.00157 

0.99857  0.00142 

0-3 

0.98823  0.00243 

0.99840  O.OO22O 

0.99855  0.00199 

0.99869  0.00180 

0.99881  0.00163 

0-35 

0.99863  0.00268 

0.99876  O.O0242 

0.99888  O.OO2I9 

0.99899  0.00198 

0.99908  0.00179 

0.4 

0.99907  O.O0286 

0.99916  O.OO259 

0.99924  O.OO234 

O.9993I  0.00212 

0.99938  0.00192 

0-45 

0.99953  0.00297 

O.99957  O.O0269 

0.99961  O.OO243 

0.99965  O.OO22O 

0.99968  O.OO2OO 

0-5 

.OOOOO  O.OO3OI 

i.ooooo  0.00272 

i.ooooo  0.00246 

i.ooooo  0.00223 

I.OOOOO  O.002O2 

0-55 

.00047  O.O0297 

1.00042  0.00269 

1.00038  0.00243 

I.OOO35  O.OO220 

I.OO03I  O.OO2OO 

0.6 

.OOO9I  O.OO286 

1.00084  O.O0259 

1.00076  0.00234 

1.00069  O.OO2I2 

I.OOO62  O.OOI92 

0.65 

.00136  O.OO268 

I.OOI23  O.OO243 

I.OOII2  O.OO22O 

i.ooioi  0.00199 

I.OOOgi  O.OOlSo 

0.7 

.OOI77  O.00244 

I.OOI6O  O.OO22O 

I.OOI45  O.OOI99 

I.OOI3I  O.OOlSo 

1.00118  0.00163 

0.75 

I.OO2I3  O.OO2I3 

I.OOI92  O.OOI93 

I.OOI74  O.OOI74 

.OOI58  O.OOI58 

I.OOI43  O.OOI43 

0.8 

I.OO244  O.OOI77 

I.OO222  O.OOl6o 

I.OOI99  O.OOI45 

.OOlSo  O.OOI3I 

1.00163  O.OOIig 

0.85 

1.00268  0.00137 

I.OO242  O.OOI24 

I.OO22O  O.OOII2 

.OOI99  O.OOIO3 

I.OOlSo  0.00092 

0.9 

I.OO286  O.OOO93 

1.00260  0.00084 

I.OO234  O.OOO76 

.OO2I2  O.OOO69 

I.OOI92  0.00062 

o-9S 

I.OO298  0.00047 

1.00269  O.OOO43 

I.OO243  O.OOO39 

.OO22O  O.OOO35 

I.OO20O  O.00032 

I.O 

1.00301  o.oo 

1.00273  o.oo 

1.00246  o.oo 

.OO223  °-°° 

.00202  o.oo 

1.05 

1.00298  0.00047 

1.00269  O.OOO43 

I.OO243  O.OOO39 

.OO22O  0.00035 

.00200  0.00032 

I.I 

1.00286  0.00093 

I.0026O  0.00084 

I.OO234  O.OO076 

.OO2I2  O.OOO69 

.00192  0.00062 

1-15 

1.00268  0.00137 

I.OO242  O.OOI24 

I.OO22O  O.OOII2 

I.OOI99  O.OOIO3 

.00180  0.00092 

1.2 

1.00244  0.00177 

I.OO222  O.OOI6O 

1.00199  O.OOI45 

I.OOlSo  O.OOI3I 

.00163  0.00119 

1-25 

1.00213  0.00213 

I.OOI92  O.OOI93 

I.OOI74  O.OOI74 

I.OOI58  O.OOI58 

.00143  0.00143 

'•3 

1.00177  0.00244 

I.OOl6o  O.0022O 

I.OOI45  O.OOI99 

I.OOI3I  O.OOlSo 

.00118  0.00163 

i-35 

1.00136  0.00268 

I.OOI23  O.O0243 

I.OOII2  O.OO22O 

i.ooioi  0.00199 

.00091  0.00180 

1.4 

1.00091  0.00286 

1.00084  O.O0259 

1.00076  O.OO234 

1.00069  O.OO2I2 

.00062  0.00192 

i  -45 

1.00047  0.00297 

I.OOO42  O.OO269 

1.00038  O.OO243 

I.OOO35  O.OO22O 

.OOO3I  O.OO2OO 

i-5 

i.  ooooo  0.00301 

i.ooooo  0.00272 

i.ooooo  0.00246 

i.ooooo  0.00223 

I.OOOOO  O.OO2O2 

i-SS 

0.99953  0.00297 

O.99957  O.OO269 

0.99961  O.OO243 

0.99965  O.OO22O 

0.99968  O.OO2OO 

1.6 

0.99907  0.00286 

0.99916  O.OO259 

0.99924  O.OO234 

0.99931  O.OO2I2 

0.99938  O.OOI92 

1.65 

0.99863  0.00268 

0.99876  O.OO242 

0.99888  O.OO2I9 

0.99899  O.OOI98 

0.99908  O.OOI79 

i-7 

0.99823  0.00243 

0.99840  O.OO22O 

0.99855  O.OOI99 

0.99869  O.OOlSo 

0.99881  0.00163 

1-75 

0.99787  O.O02I2 

0.99808  O.OOI92 

0.99826  O.OOI74 

0.99843  O.OOI57 

0.99857  O.OOI42 

1.8 

O-99757  O.OOI76 

0.99780  O.OOI6O 

0.99801  O.OOI44 

0.99820  O.OOI3I 

0.99837  O.OOIlS 

1.85 

O.99732  O.OOI36 

0.99758  O.OOI23 

O.9978l  O.OOII2 

0.99802  o.ooioi 

0.99821  O.OOO9I 

i.  9 

0.99715  0.00093 

O.99742  O.O0084 

0.99766  O.OO076 

0.99788  O.O0069 

0.99809  O.OOO62 

i-9S 

0.99704  O.O0047 

O.99732  O.O0042 

O.99757  O.O0038 

0.99780  0.00035 

0.99801  O.O003I 

2.O  O.997OO     O.OO 


0.99728   o.oo 


0.99754   o.oo 


0-99777    °'°° 


0.99799   o.oo 


Note.  Negative  quantities  are  in  heavy  type. 

Examples,     tanh  (3.25  +  to. 75)  =  1.00213  +  i 0.00213. 
tanh  (3.30  +  »  1.50)  =  i.ooooo  —  i  0.00272. 


[87] 


TABLE  IX      HYPERBOLIC  TANGENTS,     tanh  (*  +  iq)  =u  +  iv.     CONTINUED 


-  3-50 


=  3-55 


x  =  3.60 


x  =  3.70 


0 

0.05 

O.I 

0.15 

O.2 

0.99818 
0.99820 
0.99827 
0.99837 
0.99853 

o.oo 
0.00028 
0.00056 
0.00083 
0.00107 

0.99835 
0.99837 
0.99843 

0-99853 
0.99867 

o.oo 
0.00026 
0.00051 
0.00075 
0.00097 

0.99851 

0.99853 
0.99858 

0.99867 

0.99879 

o.oo 
0.00023 
0.00046 
0.00068 
0.00088 

0.99865 

0.99867 
0.99872 
0.99880 
0.99891 

O.OO 
O.OOO2I 
0.00042 
O.OOo6l 
O.OOO79 

0.99878 
0.99879 
0.99884 
0.99891 
0.99901 

o.oo 

0.00019 
0.00038 

0.00055 
0.00072 

0.25 

o-3 

0.35 
0.4 

0.45 

0.99871 
0.99893 
0.99917 
0.99944 
0.99971 

0.00129 
0.00147 
0.00162 

0.00173 

0.00180 

0.99883 
0.99903 
0.99925 
0.99949 
0.99974 

0.00117 
0.00133 
0.00147 
0.00157 
0.00163 

0.99894 

0.99912 

0.99932 
0.99954 
0.99977 

0.00105 

O.OOI2I 

0.00133 

0.00142 
0.00147 

0.99904 

0.99921 
0-99939 

0.99958 

0-99979 

0.00095 
O.OOI09 
O.OOI2O 
0.00128 
0.00133 

0.99914 
0.99928 
0.99944 
0.99962 
0.99981 

0.00086 
0.00099 
0.00109 
0.00116 

O.OOI2I 

o-S 
o-5S 
0.6 
0.65 
0.7 

I.OOOOO 

I  00028 
1.00056 
1.00083 
1.00107 

0.00182 
0.00180 
0.00174 
0.00163 
0.00148 

I.OOOOO 

1.00026 

1.00051 
1.00075 
1.00097 

0.00165 
0.00163 
0.00157 
0.00147 
0.00134 

I.OOOOO 

1.00025 
1.00048 
1.00068 

1.00088 

0.00149 
0.00148 
0.00142 

0.00133 

O.OOI2I 

I.OOOOO 
I.OO02I 
I.OOO42 
I.  OOO6  1 
I.OOO79 

0.00135 
0.00133 
O.OOI29 
O.OO  I  2O 

0.00109 

I.OOOOO 

1.00019 
1.00038 
1.00055 
1.00072 

O.OOI22 
O.OOI2I 

0.00116 
0.00109 
0.00099 

o.75 
0.8 
0.85 
0.9 
0-95 

1.00129 
1.00148 
1.00163 
1.00174 
1.00180 

0.00129 
0.00107 
0.00083 
0.00056 
0.00029 

1.00117 
1.00133 

1.00147 

1.00157 

1.00163 

0.00117 
0.00097 
0.00075 
0.00051 
0.00026 

1.00106 

I.OOI2I 

1.00133 

1.00142 
1.00148 

0.00106 
0.00088 
0.00068 
0.00046 
0.00023 

I.OOO96 
I.OOI08 
I.OOI2O 
I.OOI29 
I.OOI34 

0.00096 
0.00080 
0.00061 
0.00042 

O.OOO2I 

1.00086 
1.00099 
1.00109 
1.00116 

I.OOI2I 

0.00086 
0.00072 
0.00056 
0.00038 
0.00019 

I.O 

1.05 
i.i 
i-iS 

1.2 

1.00183 
1.00180 
1.00174 
1.00163 
1.00148 

0.00 

0.00029 
0.00056 
0.00083 
0.00107 

1.00165 
1.00163 

1.00157 

1.00147 

1.00133 

o.oo 
0.00026 
0.00051 
0.00075 
0.00097 

1.00149 
1.00148 
1.00142 

1.00133 

I.OOI2I 

0.00 

0.00023 
0.00046 
0.00068 
0.00088 

I.OOI35 
I.OOI34 
1.00129 
I.OOI2O 
I.OOI08 

O.OO 
O.OOO2I 
O.OOO42 
O.OOO6l 
O.OOO8O 

I.OOI22 
I.OOI2I 

1.00116 
1.00109 
1  .00099 

o.oo 
0.00019 
0.00038 
0.00056 
0.00072 

1-25 

i-3 
i-35 
1.4 

1-45 

1.00129 
1.00107 
1.00083 
1.00056 
1.00028 

0.00129 
0.00148 
0.00163 
0.00174 
0.00180 

1.00117 
1.00097 

1.00075 

1.00051 
1.00026 

0.00117 
0.00134 
0.00147 
0.00157 
0.00163 

1.00106 

1.00088 

1.00068 
1.00048 
1.00025 

0.00106 

O.OOI2I 

0.00133 
0.00142 
0.00148 

I.OOO96 
I.OOO79 
1.  000(5  1 
I.OOO42 
I.OO02I 

O.OOO96 
O.OOIO9 
O.OOI2O 
O.OOI29 
O.OOI33 

1  .00086 
1.00072 

1.00055 
1.00038 
1.00019 

0.00086 
0.00099 
0.00109 
o.  00116 

O.OOI2I 

i-5 
I-5S 
1.6 
1.65 
i-7 

I.OOOOO 

0.99971 
0.99944 
0.99917 
0.99893 

0.00182 
0.00180 
0.00173 
0.00162 
0.00147 

I.OOOOO 

0.99974 
0.99949 
0.99925 
0.99903 

0.00165 
0.00163 
0.00157 
0.00147 
0.00133 

I.OOOOO 

0.99977 
0.99954 
0.99932 

0.99912 

0.00149 
0.00147 
0.00142 
0.00133 

O.OOI2I 

I.OOOOO 
0.99979 
0.99958 
0.99939 
0.99921 

0.00135 
0.00133 
O.OOI28 
O.OOI20 
O.OOIO9 

I.OOOOO 

0.99981 
0.99962 
0-99944 
0.99928 

O.OOI22 
O.OOI2I 

o.  00116 

O.OOIO9 
O.OOO99 

1-75 
1.8 
1.85 
1.9 

i-95 

0.99871 

0.99853 
0.99837 

0.99827 
0.99820 

0.00129 
0.00107 
0.00083 
0.00056 
0.00028 

0.99883 

0.99867 

0-99853 
0.99843 
0.99837 

0.00117 
0.00097 
0.00075 
0.00051 
0.00026 

0.99894 
0.99879 

0.99867 
0.99858 
0.99853 

0.00105 
0.00088 
0.00068 
0.00046 
0.00023 

0.99904 
0.99891 
0.99880 
0.99872 
0.99867 

O.OOO95 
O.OOO79 
0.00061 
O.OOO42 
O.OO02I 

0.99914 
0.99901 
0.99891 
0.99884 
0.99879 

O.OO086 
O.OOO72 
0.00055 
O.OOO38 
O.OOOI9 

2.O 

0.99818 

O.OO 

0.99835 

o.oo 

0.99851 

o.oo 

0.99865 

O.OO 

0.99878 

0.00 

Note. 

Negative  quantities  are  in  heavy  type. 

Examples,  tanh  (3.60  + 

i  0.80)  = 

I.OOI2I  +  i  O.OOO88. 

tanh  (3.70  -+•  *  1.70)  =  0.99928  —  i  0.00099. 


[88] 


TABLE  IX.     HYPERBOLIC  TANGENTS,     tanh  (x  +  iq)  =  u  +  iv.     CONTINUED 


=  3-75 


x  =  3.80 


=  3-85 


3-90 


=  3-95 


o 

0.99889 

o.oo 

0.99900 

o.oo 

0.99909 

o.oo 

0.99918 

o.oo 

0.99926 

o.oo 

0.05 

0.99891 

0.00017 

0.99901 

0.00016 

0.99911 

0.00014 

0.99919 

0.00013 

0.99927 

O.OOOI2 

O.I 

0.99895 

0.00034 

0.99905 

0.00031 

0.99914 

0.00028 

0.99922 

0.00025 

0.99930 

0.00023 

0.15 

0.99901 

0.00050 

0.99911 

0.00045 

0.99919 

0.00041 

0.99927 

0.00037 

0.99934 

0.00034 

0.2 

0.99911 

0.00065 

0.99919 

0.00059 

0.99927 

0.00053 

0.99934 

0.00048 

0.99940 

0.00044 

0.25 

0.99922 

0.00078 

0.99929 

0.00071 

0.99936 

0.00064 

0.99942 

0.00058 

0.99948 

0.00052 

°-3 

0-99935 

0.00089 

0.99941 

0.00081 

0.99947 

0.00073 

0.99952 

0.00066 

0.99956 

0.00060 

0-35 

0.99950 

0.00099 

0-99955 

0.00089 

0.99959 

0.00081 

0.99963 

0.00073 

0.99966 

0.00066 

0.4 

0.99966 

0.00105 

0.99969 

0.00095 

0.99972 

0.00086 

0.99975 

0.00078 

0.99977 

0.00071 

0-45 

0.99983 

0.00109 

0.99984 

0.00099 

0.99986 

0.00089 

0.99987 

0.00081 

0.99988 

0.00073 

o.S 

1.  00000 

O.OOIII 

I.OOOOO 

O.OOIOO 

I.OOOOO 

0.00091 

I.OOOOO 

0.00082 

I.OOOOO 

0.00074 

0-55 

1.00017 

0.00109 

1.00016 

0.00099 

1.00014 

0.00089 

1.00013 

0.00081 

I.OOOI2 

0.00073 

0.6 

1.00034 

0.00105 

1.00031 

0.00095 

1.00028 

0.00086 

1.00025 

0.00078 

1.00023 

0.00071 

0.65 

1.00050 

0.00099 

1.00045 

0.00089 

1.00041 

0.00081 

1.00037 

0.00073 

1.00034 

0.00066 

0.7 

1.00065 

0.00090 

1.00059 

0.00081 

1.00053 

0.00073 

1.00048 

0.00066 

1.00044 

0.00060 

0-75 

1.00078 

0.00078 

1.00071 

0.00071 

1.00064 

0.00064 

1.00058 

0.00058 

1.00052 

0.00052 

0.8 

1.00089 

0.00065 

1.00081 

0.00059 

1.00073 

0.00053 

1.00066 

0.00048 

1.00060 

0.00044 

0.85 

1.00099 

0.00050 

1.00089 

0.00045 

1.00081 

0.00041 

1.00073 

0.00037 

1.00066 

0.00034 

0.9 

1.00105 

0.00034 

1.00096 

0.00031 

1.00086 

0.00028 

1.00078 

0.00025 

1.00071 

0.00023 

o-9S 

1.00109 

0.00017 

1.00099 

0.00016 

1.00089 

0.00014 

1.00081 

0.00013 

1.00073 

O.OOOI2 

I.O 

I.OOIII 

o.oo 

I.OOIOO 

o.oo 

1.00090 

o.oo 

1.00082 

o.oo 

1.00074 

O.OO 

1.05 

1.00109 

0.00017 

1.00099 

0.00016 

1.00089 

0.00014 

1.00081 

0.00013 

1.00073 

0.00012 

i.i 

1.00105 

0.00034 

1.00096 

0.00031 

1.00086 

0.00028 

1.00078 

0.00025 

1.00071 

O.OO023 

i-iS 

1.00099 

0.00050 

1.00089 

0.00045 

1.00081 

0.00041 

1.00073 

0.00037 

1.00066 

O.OOO34 

1.2 

1.00089 

0.00065 

1.00081 

0.00059 

1.00073 

0.00053 

1.00066 

0.00048 

1.00060 

O.OOO44 

I.2S 

1.00078 

0.00078 

1.00071 

0.00071 

1.00064 

0.00064 

1.00058 

0.00058 

1.00052 

O.OO052 

i-3 

1.00065 

0.00090 

1.00059 

0.00081 

1.00053 

0.00073 

1.00048 

0.00066 

1.00044 

O.OO06O 

i-35 

1.00050 

0.00099 

1.00045 

0.00089 

1.00041 

0.00081 

1.00037 

0.00073 

1.00034 

O.OO066 

1.4 

1.00034 

0.00105 

1.00031 

0.00095 

1.00028 

0.00086 

1.00025 

0.00078 

1.00023 

O.OOO7I 

1-45 

1.00017 

0.00109 

1.00016 

0.00099 

1.00014 

0.00089 

1.00013 

0.00081 

I.OOOI2 

O.OOO73 

i-5 

I.OOOOO 

O.OOIII 

I.OOOOO 

O.OOIOO 

I.OOOOO 

0.00091 

I.OOOOO 

0.00082 

I.OOOOO 

O.OO074 

i-SS 

0.99983 

o.ooiog 

0.99984 

0.00099 

0.99986 

0.00089 

0.99987 

0.00081 

0.99988 

O.00073 

1.6 

0.99966 

0.00105 

0.99969 

0.00095 

0.99972 

0.00086 

0-99975 

0.00078 

0.99977 

O.OOO7I 

1.65 

0.99950 

0.00099 

0.99955 

00089 

0.99959 

0.00081 

0.99963 

0.00073 

0.99966 

O.OOO66 

i-7 

0-99935 

0.00089 

0.99941 

0.00081 

0.99947 

0.00073 

0-99952 

0.00066 

0.99956 

O.OOO6O 

i-75 

0.99922 

0.00078 

0.99929 

0.00071 

0.99936 

0.00064 

0.99942 

0.00058 

0.99948 

O.OOO52 

1.8 

0.99911 

0.00065 

0.99919 

0.00059 

0.99927 

0.00053 

0.99934 

0.00048 

0.99940 

O.OOO44 

1.85 

0.99901 

0.00050 

0.99911 

0.00045 

0.99919 

0.00041 

0.99927 

0.00037 

0.99934 

O.OOO34 

1.9 

0.99895 

0.00034 

0.99905 

0.00031 

0.99914 

0.00028 

0.99922 

0.00025 

0.99930 

O.OOO23 

J-95 

0.99891 

0.00017 

0.99901 

0.00016 

0.99911 

0.00014 

0.99919 

0.00013 

0.99927 

O.OOOI2 

2.O  0.99889     O.OO 


0.99900   o.oo 


0.99909   o.oo 


0.99918   o.oo 


0.99926   o.oo 


Note.  Negative  quantities  are  in  heavy  type. 

Examples,     tanh  (3.95  +  i  0.95)  =  1.00073  +  »  0.00012. 
tanh  (3.95  -f  i  1.05)  =  1.00073  —  i  0.00012. 


[89] 


TABLE  X.    HYPERBOLIC   SINES,     sinh  (x  +  iq)  =  r  /y_ 


x  =  0.05 


x  =  0.15 


o 

o.ooo 

9° 

0.05002 

o.ooo 

0.10017 

o.ooo 

0.15056 

o.ooo 

0.20134 

o.ooo 

0.05 

0.07846 

90 

0.09305 

57-593 

0.12724 

38.300 

0.16978 

27.861 

0.21608 

21.739 

O.I 

0.15643 

90 

0.16424 

72.493 

0.18576 

57-819 

0.21712 

46.771 

0.25497 

38.746 

0.15 

0-23345 

9° 

0.23874 

78-245 

0.25403 

67-455 

0.27779 

58.195 

0.30827 

50.576 

O.2 

0.30902 

90 

0.31304 

81.259 

0.32485 

72.947 

0-34375 

65.382 

0.36882 

58.723 

O.25 

0.38268 

9° 

0.38594 

83-123 

0.39558 

76.471 

0.41124 

70.229 

0.43422 

64.522 

°-3 

0.45399 

90 

0.45672 

84.400 

0.46491 

78.932 

0.47831 

73-7" 

0.49663 

68.825 

0.35 

0.52250 

90 

0.52487 

85-339 

0.53201 

80.762 

0.54376 

76-344 

0-55995 

72.147 

0.4 

0.58778 

9° 

0.58989 

86.066 

0.59626 

82.189 

0.60676 

78.419 

0.62131 

74.802 

0.45 

0.64944 

90 

0-65135 

86.652 

0.65713 

83.344 

0.66667 

80.111 

0.67994 

76.988 

°-5 

0.70711 

90 

0.70803 

87-I39 

0.71417 

84.308 

0.72296 

81.532 

0.73521 

78.835 

o-55 

0.76041 

90 

0.76202 

87-557 

0.76698 

85-134 

0.77527 

82.753 

0.78661 

80.431 

0.6 

0.80902 

00 

0.81053 

87.921 

0.81520 

85.858 

0.82291 

83.826 

0.83370 

81.839 

0.65 

0.85264 

90 

0.85407 

88.246 

0.85851 

86.505 

0.86583 

84.787 

0.87609 

83.103 

0.7 

0.89101 

90 

0.89237 

88.542 

0.89662 

87.093 

0.90364 

85.662 

0.91347 

84.257 

o-7S 

0.92388 

90 

0.92523 

88.814 

0.92930 

87.636 

0.93607 

86.471 

0.94556 

85.326 

0.8 

0.95106 

90 

0.95237 

89.070 

0.95632 

88.145 

0.96290 

87-231 

0.97213 

86.331 

0.85 

0.97237 

90 

0.97366 

89-313 

0.97752 

88.629 

0.98396 

87-953 

0.99300 

87.287 

0.9 

0.98769 

90 

0.98895 

89-547 

0.99275 

89.095 

0.99909 

88.649 

1.00800 

88.209 

0-95 

0.99692 

90 

0.99817 

89-775 

1.00194 

89.550 

1.00822 

89.329 

1.01704 

89.110 

I.O 

I.OOOOO 

90 

1.00125 

90.000 

1.00500 

90.000 

1.01127 

90.000 

1.02007 

90.000 

1.05 

0.99692 

90 

0.99817 

90.225 

1.00194 

90.450 

1.00822 

90.671 

1.01704 

90.890 

i.i 

0.98769 

90 

0.98895 

90-453 

0.99275 

90.905 

0.99909 

9i-35i 

1.00800 

91.791 

"5 

0.97237 

90 

0.97366 

90.687 

0.97752 

9*-37i 

0.98396 

92.047 

0.99300 

92.713 

1.2 

0.95106 

90 

0.95237 

90.930 

0.95632 

9I-855 

0.96290 

92.769 

0.97213 

93.669 

I-2S 

0.92388 

90 

0.92523 

91.186 

0.92930 

92-364 

0.93607 

93-529 

0.94556 

94.674 

i-3 

0.89101 

90 

0.89237 

91.458 

0.89662 

92.907 

0.90364 

94.338 

0.91347 

95-743 

i-35 

0.85264 

90 

0.85407 

91-754 

0.85851 

93-495 

0.86583 

95-213 

0.87609 

96.897 

1.4 

0.80902 

90 

0.81053 

92.079 

0.81520 

94.142 

0.82291 

96.174 

0.83370 

98.161 

i  -45 

0.76041 

90 

0.76202 

92.443 

0.76698 

94.866 

0.77527 

97.247 

0.78661 

99.569 

1-5 

0.70711 

90 

0.70803 

92.860 

0.71417 

95.692 

0.72296 

98.468 

0.73521 

101.165 

1-55 

0.64944 

90 

0.65135 

93-348 

0.65713 

96.656 

0.66667 

99.889 

0.67994 

103.012 

1.6 

0.58778 

90 

0.58989 

93-934 

0.59626 

97.811 

0.60676 

101.581 

0.62131 

105.198 

1.65 

0.52250 

90 

0.52487 

94.661 

0.53201 

99-238 

0.54376 

103.656 

0-55995 

107-853 

i-7 

0.45399 

90 

0.45672 

95.600 

0.46491 

101.068 

0.47831 

106.289 

0.49663 

111.175 

i-7S 

0.38268 

90 

0.38594 

96.877 

0.39558 

103.529 

0.41124 

109.771 

0.43242 

115.478 

1.8 

0.30902 

90 

0.31304 

98.741 

0.32485 

107.053 

0-34375 

114.618 

0.36882 

121.277 

1.85 

0.23345 

90 

0.23874 

101.755 

0.25403 

112.545 

0.27779 

121.805 

0.30827 

129.424 

1.9 

0.15643 

90 

0.16424 

107.507 

0.18576 

122.181 

0.21712 

133.229 

0.25497 

141.254 

I-  95 

0.07846 

90 

0.09305 

122.407 

0.12724 

141.700 

0.16978 

152-139 

0.21608 

158.261 

2.O 

o.oo 

90 

0.05002 

180.000 

0.10017 

180.000 

0.15056 

180.000 

0.20134 

180.000 

Example.      sinh  (0.15  +  to.  15)  =  0.27779 


=  0.27779  /58°.ii/.42*. 


TABLE  X.    HYPERBOLIC  SINES,     sinh  (at  +  iq)  =  r  /T.     CONTINUED 


=  0.25 


x  =  0.3 


=  0.35 


*  =  0.4 


x  =  0.45 


0.25261 

o.ooo 

0.30452 

0.000 

0.35719 

o.ooo 

0.41075 

o.ooo 

0.46534 

o.ooo 

0.26452 

17.814 

0.31447 

15.118 

0.36571 

13.169 

0.41818 

11.703 

0.47191 

10.567 

0.29782 

32.890 

0.34235 

28.533 

0.38994 

25.214 

0-43953 

22.629 

0.49093 

20.576 

0.34396 

44-943 

0.38370 

39-493 

0.42671 

35.516 

0.47246 

32.288 

0.52062 

29.642 

o.399J3 

52.992 

0.43385 

48.122 

0.47231 

44.008 

0.51401 

40.536 

0.55860 

37.601 

0.45854 

59-405 

0.48906 

54.882 

0.52348 

50.921 

0.56140 

47.471 

0.60249 

44-473 

0.51954 

64.327 

0.54666 

60.242 

0.57766 

56.843 

0.61223 

53-288 

0.65012 

50-374 

0.58036 

68.215 

0.60476 

64-575 

0.63292 

61.237 

0.66462 

58.200 

0.69968 

55-454 

0.63977 

7I-37I 

0.66199 

68.151 

0.68781 

65.157 

0.71708 

62.393 

0.74969 

59-856 

0.69685 

73-999 

0.71730 

71.166 

0.74II9 

68.503 

0.76844 

66.018 

0.79895 

63.712 

0.75088 

76.238 

0.76989 

73-759 

0.79220 

71.408 

0.81775 

69.196 

0.84649 

67.125 

0.80127 

78.185 

0.81912 

76.028 

O.840I2 

73.971 

0.86426 

72.021 

0.89149 

70.184 

0.84754 

79.911 

0.86443 

78.049 

0.88436 

76.267 

0.90732 

74.568 

0-93330 

72.958 

0.88927 

81.464 

0-90539 

79-878 

0.92444 

78.353 

0.94642 

76.893 

0.97136 

75.504 

0.92612 

82.887 

0.94161 

8i.557 

0.95994 

80.274 

0.98113 

79.043 

1.00521 

77.868 

0-95779 

84.207 

0.97277 

83.119 

0.99052 

82.068 

1.01107 

81.056 

1.03446 

80.087 

0.98403 

85-450 

0.99862 

84-593 

I.OI592 

83.762 

1.03597 

82.962 

1.05880 

82.194 

1.00464 

86.635 

1.01894 

85.999 

1.03590 

85.383 

1-05557 

84.788 

1.07798 

84.216 

1.01949 

87-779 

1-03357 

87-358 

I.05O29 

86.951 

1.06970 

86.556 

1.09182 

86.177 

1.02843 

88.896 

1.04239 

88.687 

1.05898 

88.484 

1.07822 

88.287 

1.10017 

88.098 

1.03141 

90.000 

1-04534 

90.000 

i.  06  1  88 

90.000 

1.08107 

90.000 

1.10297 

90.000 

1.02843 

91.104 

1.04239 

91.313 

1.05898 

91.516 

1.07822 

91-713 

1.10017 

91.902 

1.01949 

92.221 

1-03357 

92.642 

1.05029 

93.050 

1.06970 

93-444 

1.09182 

93-823 

i  .00464 

93-365 

1.01894 

94.001 

1.03590 

94.617 

1-05557 

95-212 

1.07798 

0.98403 

94-550 

0.99862 

95.407 

1.01592 

96.238 

1-03597 

97.038 

1.05880 

97.806 

0-95779 

95-793 

0.97277 

96.881 

0.99052 

97.932 

1.01107 

98.944 

1.03446 

99-9J3 

0.92612 

0.94161 

98.443 

0.95994 

99.726 

0.98113 

100.957 

1.00521 

102.132 

0.88927 

98*536 

0-90539 

IOO.I22 

0.92444 

101.647 

0.94642 

103.107 

0.97136 

104.496 

0-84754 

100.090 

0.86443 

IOI.95I 

0.88436 

103.733 

0.90732 

105.432 

0-93330 

107.042 

0.80127 

101.815 

0.81912 

103.972 

0.84012 

106.029 

0.86426 

107.979 

0.89149 

109.816 

0.75088 

103.762 

0.76989 

IO6.24I 

0.79220 

108.592 

0.81775 

110.804 

0.84649 

112.875 

0.69685 

106.001 

0.71730 

108.834 

0.74119 

111.497 

0.76844 

113.982 

0.79895 

116.288 

0.63977 

108.629 

0.66199 

III.849 

0.68781 

114.843 

0.71708 

117.607 

0.74969 

120.144 

0.58036 

111.785 

0.60476 

115425 

0.63292 

118.763 

0.66462 

121.800 

0.69968 

124.546 

0.51954 

0.54666 

119.758 

0.57766 

123.157 

0.61223 

126.712 

0.65012 

129.626 

0-45854 

120.595 

0.48906 

I25.II8 

0.52348 

129.079 

0.56140 

132.529 

0.60249 

I35.527 

0.39913 

127.008 

0-43385 

131.878 

0.47231 

135.992 

0.51401 

139.464 

0.55860 

142.399 

0.34396 

I35-057 

>  0.38370 

140.507 

0.42671 

144.484 

0.47246 

147.712 

0.52062 

150.358 

0.29782 

147.110 

0.34235 

151.467 

0.38994 

154.786 

0-43953 

I57-37I 

0.49093 

159.424 

0.26452 

162.186 

0.31447 

164.882 

0.36571 

166.831 

0.41818 

168.297 

0.47191 

169.433 

0.25261 

180.000 

0.30452 

iSo.OOO 

0.35719 

180.000 

0.41075 

180.000 

0.46534 

180.000 

Example.      sinh  (0.40  +  £0.25)  =  0.56140  ( 


=  0.56140  /47°.28'.i6". 


[90 


TABLE  X.    HYPERBOLIC  SINES,     sinh  (x  +  iq)  =  r 


CONTINUED 


x  =  0.5 


=  °-55 


x  =  0.6 


*  =  0.65 


0 

0.05 

O.I 

0.15 

0.2 

0.52110  o.ooo 

0.52697  9.665 
0.54407  18.918 
0.57100  27.453 
0.60583  35.111 

0.57815       o.ooo 

0.58345      8.936 
0.59894    17.559 
0.62350     25.625 
0.65555    32.990 

0.63665  o.ooo 

0.64147  8.337 
0.65559  16.432 

0.67810  24.086 

0.70769  31.174 

0.69675       o.ooo 

0.70115      7.839 
0.71409    15.486 

0.73482      22.781 
0.76220      29.613 

0.75858  o.ooo 
0.76263  7.419 
0.77455  14.685 
0.79369  21.665 
0.81911  28.263 

0.25 

o-3 
0-35 
0.4 

0-45 

0.64652  41.871 

0.69112  47-793 
0-73793  52-980 
0.78552  57-542 
0.83266  61.584 

0.69333    39-610 
0.73510    45.511 
0.77927    50.759 

0.82447      55-437 
0.86951      59.628 

0.74282  37.642 

0.78194  43-494 
0.82361  48.769 
0.86650  53-529 
0.90946  57.838 

0.79492    35.926 

0.83160     41.710 
0.87090     46.989 

0.91156    51.803 
0.95249    56.204 

0.84965  34.426 
0.88406  40.133 
0.92112  45-397 
0.95966  50.245 
0.99861  54.716 

0-55 
0.6 
0.65 
0.7 

0.87837  65.198 
0.92182  68.462 
0.96232  71-441 
0.99927  74-189 
1.03220  76-751 

0.91338     63.411 
0.95524     66.854 
0-99437      70.016 
1.03004      72.948 
1.06214      75.693 

0.95149  61.762 
0.99171  65.360 
1.02948  68.685 
1.06411  71.784 
1.09509  74.696 

0.99270     60.245 

.03134    63.976 
.06769    67.445 

.101  1  1      70.694 

.13108    73.760 

1.03704  58.853 
1.07409  62.698 
1.10904  66.294 
1.14125  69.677 
1.17019  72.884 

0-75 
0.8 
0.85 
0.9 
0-95 

1.06070  79.164 
1.08446  81.461 
1.10320  83.669 
1.11672  85.814 
1.12489  87.917 

1.08987      78.287 
1.11300      80.763 
1.13127      83.148 
1.14446      85.467 
1.15244      87.744 

1.  12200  77-459 
1.14448  8O.IO2 
1.16226  82.653 
I.I75IO  85.121 
I.I8287  87.580 

.15715    76.678 

.17892      79-477 
.19623      82.185 
.20871      84.826 
1.21626      87.424 

1.19541  75-946 
1.21653  78.890 
1.23327  81.744 
1.24538  84.532 
1.25271  87.277 

I.O 

.1 
•IS 

.2 

1.12763  90.000 
1.12489  92.083 
1.11672  94.186 
1.10320  96.331 
1.08446  98.539 

1.15510     90.000 
1.15244     92.256  ( 
1.14446     94-533 
1.13127      96.852 
1.11300     99.237 

1.18547  9O.OOO 
1.18287  92.420 
I.I75IO  94.879 
I.I6226  97-347 
1.14448  99.898 

1.21879      90.000 
1.21626     92.576 
1.20871      95.174 
1.19623      97.815 
1.17892    100.523 

1.25517  90.000 
1.25271  92.723 
1.24538  95-468 
1.23327  98.256 
1.21653  ioi.no 

•25 

•3 
•35 
1.4 

1-45 

1.06070  100.836 
1.03220  103.249 
0.99927  105.811 
0.96232  108.559 
0.92182  111.538 

1.08987    101.713 
1.06214    104.307 
1.03004    107.052 
0.99437    109.984 
0.95524    113.146 

I.I22OO  102.541 
1.09509  105.304 
I.O64H  IO8.2I6 
1.02948  III.3I5 
0.99174  114.640 

1.15715    103.322 
1.13108    106.240 
i.ioni    109.306 
1.06769    112.555 
1.03134    116.024 

1.19541  104.054 

1.17019  107.116 
1.14125  110.323 
1.10904  113.706 
1.07409  117.302 

i-SS 
1.6 
1.65 

0.87837  114.803 
0.83266  118.416 
0.78552  122.458 
0.73793  127.020 
0.69112  132.207 

0.91338    116.589 
0.86951    120.372 
0.82447    124.563 
0.77927    129.241 
0.73510    134-489 

0.95149  II8.238 
0.90946  I22.I62 
0.86650  126.471 
0.82361  131.231 
0.78194  136.506 

0.99270    119.755 
0.95249    123.796 
0.91156    128.197 
0.87090    133.011 
0.83160    138.290 

1.03704  121.148 
0.99861  125.284 

0.95966  129.755 
0.92112  134.603 
0.88406  139.867 

1.8 
1.85 
1.9 

0.64652  138.129 
0.60583  144.889 
0.57100  152.547 
0.54407  161.082 
0.52697  170.335 

0-69333    140-390 
0.65555    147-010 
0.62350    154-375 
0.59894    162.441 
0.58345    171.064 

0.74282  142.358 
0.70769  148.826 
0.67810  155.914 
0.65559  163.568 
0.64147  171.663 

0.79492    144-074 
0.76220    150.387 
0.73482    157.219 
0.71409    164.514 
0.70115    172.161 

0.84965  145.574 
0.81911  151.737 
0.79369  158.335 
0.77455  165.315 
0.76263  172.581 

2.0 

0.52110  180.000 

0.57815    180.000 

0.63665  iSo.OOO 

0.69675    180.000 

0.75858  180.000 

Example.      sinh  (0.70  +  i  1.70)  =  0.88406 


=  0.88406  / 130°. $2'. 01*. 


[92] 


TABLE  X.    HYPERBOLIC   SINES,     sinh  (x  +  iq)  =  r  /y.     CONTINUED 


x  = 

0-75 

x  = 

0.8 

x  = 

0.85 

x  = 

0.9 

x  = 

0.95 

9 

r 

7 

r 

y 

r 

y 

r 

y 

r 

y 

0 

o 

0 

o 

o 

o.oo 

0.82232 

o.ooo 

0.88811 

0.000 

0.95612 

O.OOO 

1.02652 

o.oob 

1.09948 

O.OOO 

0.05 

0.82605 

7.064 

0.89157 

6-759 

0-95933 

6.497 

1.02951 

6.270 

1.10227 

6.073 

O.IO 

0.83706 

14.002 

0.90178 

I3-4I5 

0.96883 

12.909 

1-03837 

12.468 

1.11056 

12.085 

0.15 

0.85481 

20.706 

0.91828 

19.877 

0.98420 

19.158 

1.05273 

18.529 

1.12400 

17.980 

O.2 

0.87846 

27.093 

0.94033 

26.073 

1.00481 

25.l8l 

1.07202 

24-399 

1.14209 

23.712 

0.25 

0.90700 

33-1" 

0.96705 

3I-955 

1.02985 

30.938 

1-09553 

30-039 

1.16418 

29.245 

0-3 

0.93932 

38.737 

0.99742 

37-500 

1.05843 

36.401 

1.12243 

35-426 

1.19227 

34-557 

0.35 

0.97427 

43-974 

1.03041 

42.702 

1.08957 

4I.565 

1.15184 

40-547 

1.21732 

39-637 

0.4 

1.01079 

48.840 

1.06500 

47-574 

1.12234 

46.433 

1.18289 

45-407 

1.24674 

44.483 

Q-45 

1.04785 

53-363 

1.10024 

52.135 

I.I5583 

5I.O22 

1.21471 

50.014 

1.27697 

49.102 

0-5 

1.08453 

57-578 

1.13522 

56.414 

1.18918 

55-353 

1.24649 

54.386 

1.30723 

53-507 

o.SS 

I.I200I 

61.522 

1.16917 

60.441 

1.22163 

59-450 

1.27748 

58.543 

1.33682 

57.714 

0.6 

I-I5356 

65.228 

1.20135 

64.245 

1.25246 

63-399 

1.30700 

62.507 

1-36505 

6i.743 

0.65 

1.18457 

68.733 

1.23115 

67-857 

1.28107 

67.048 

1-33444 

66.301 

1.39136 

65.613 

0.7 

I.2I248 

72.067 

1.25802 

71.307 

1.30693 

70.602 

1.35927 

69-949 

1.41519 

69-347 

0-75 

1.23689 

75.260 

1.28152 

74.621 

I-32955 

74.026 

1.38105 

73-474 

1.43611 

72.963 

0.8 

1.25726 

78.339 

1.30124 

77-825 

1-34858 

77-344 

1-39937 

76.898 

I-4537I 

76.484 

0.85 

1.27346 

81.330 

1.31691 

80.942 

1.36369 

80.580 

I-4I395 

80.242 

1.46778 

79-929 

0.9 

1.28520 

84-255 

1.32825 

83.996 

1.37466 

83-754 

1.42452 

83-527 

1.47797 

83-316 

o-9S 

1.29230 

87-138 

I-335I3 

87.008 

1.38130 

86.887, 

1.43094 

86.773 

1.48415 

86.668 

I.O 

1.29468 

90.000 

1-33743 

90.000 

I.38353 

90.000 

I-43309 

90.000 

1.48623 

90.000 

1.05 

1.29230 

92.862 

I-335I3 

92.992 

1.38130 

93-II3 

1.43094 

93.227 

1.48415 

93-332 

i.i 

1.28520 

95-745 

1.32825 

96.004 

1.37466 

96.246 

1.42452 

96-473 

1.47797 

96.684 

i-iS 

1.27346 

98.670 

1.31691 

99.058 

1.36369 

99.420 

I-4I395 

99.758 

1.46778 

100.071 

1.2 

1.25726 

101.661 

1.30124 

102.175 

1.34858 

102.656 

1-39937 

103.102 

I-4537I 

103.516 

I.2S 

1.23689 

104.740 

1.28152 

105-379 

I.32955 

105.974 

1.38105 

106.526 

1.43611 

107.037 

i-3 

I.2I248 

107.933 

1.25802 

108.693 

1.30693 

109.398 

I-35927 

110.051 

1.41519 

110.653 

i-35 

I.I8457 

111.267 

1-23115 

112.143 

1.28107 

112.952 

1.33444 

113.699 

1.39136 

114.387 

•4 

I-I5356 

114.772 

1.20135 

"5-755 

1.25246 

116.601 

1.30700 

117-493 

1-36505 

118.257 

•45 

I.I2OOI 

118.478 

1.16917 

119-559 

1.22163 

120.550 

1.27748 

121-457 

1.33682 

122.286 

•5 

1.08453 

122.422 

1.13522 

123.586 

1.18918 

124.647 

1.24649 

125.614 

1.30723 

126.493 

•55 

1.04785 

126.637 

1.10024 

127.865 

1-15583 

128.978 

1.21471 

129.986 

.27697 

130.898 

.6 

I.OIO79 

131.160 

1.06500 

132.426 

1.12234 

I33-567 

1.18289 

134-593 

.24674 

I35-5I7 

•6S 

0.97427 

136.026 

1.03041 

137.298 

1.08957 

138.435 

1.15184 

139-453 

.21732 

140.363 

•7 

0.93932 

141.263 

0.99742 

142.500 

1.05843 

143-599 

1.12243 

144-574 

19227 

145-443 

•75 

O.907OO 

146.889 

0.96705 

148.045 

1.02985 

149.062 

1.09553 

149.961 

.16418 

150.755 

.8 

0.87846 

152.907 

0.94033 

153-927 

1.00481 

154.819 

1.07202 

155.601 

.14209 

156.288 

1.85 

0.85481 

159.294 

0.91828 

160.123 

0.98420 

160.842 

1-05273 

161.471 

.12400 

162.020 

1.9 

0.83706 

165.998 

0.90178 

166.585 

0.96883 

167.091 

1.03837 

167.532 

.11056 

167.915 

i-95 

0.82605 

172.936 

0.89157 

173.241 

0-95933 

I73-503 

1.02951 

I73.730 

.10227 

173.927 

2.0  0.82232      ISO.OOO         0.888II      iSo.OOO         0.95612      iSo.OOO          1.02652      iSo.OOO          1.09948     iSo.OOO 


Example.       sinh  (0.90  +  i  i.o)  =  1.43309  /go0. 


[93] 


TABLE  X.    HYPERBOLIC  SINES,     sinh  (*  +  iq)  =  r  /r     CONTINUED 


X  =  I.O 

x  =  1.05 

X  —  I.I 

x  =  1.15 

X  =  1.2 

9 

f      y 

f    y 

r      y 

r     y 

r      y 

0 

0.05 

O.I 

0.15 

O.2 

0 

1.17520  o.ooo 
1.17782  5.900 
1.18552  11.748 
1.19816  17.496 
1.21515  23.105 

0 

1.25386   o.ooo 

I.2563I     5.748 
1.26358   11-453 
1.27540   17.071 
I.29I37   22.568 

o 

L33565   o.ooo 
1-33795   5-615 
1.34478  11.192 
1-35590  16.694 
1.37093   22.092 

0 

1.42078   o.ooo 

1.42294  5.497 

1.42936   10.961 
1.43983  16.361 
1.45399  21.670 

o 

1.50946  o.ooo 

I.5II50  5.393 
I.5I755  10.757 
I.5274I  16.065 
1.54077  21.294 

0.25 

0-35 
0.4 

0-45 

1.23594  28.541 
1.25984  33.784 
1.28612  38.821 

1.31400  43-651 
1.34272  48.276 

I.3I095    27.915 
I-3335I   33-093 
I-35837   38.090 
1.38479   42.902 
I.4I207   47-530 

1.38939  27.359 
1.41070  32.477 
1.43421   37-435 
1.45926  42.227 
1.48517  46.855 

1.47141   26.864 
i.49x55  3I-926 
1.51381  36.847 
1.53756  41.620 
1.56217  46.245 

I.5572I  26.421 
1.57626  31.433 

1-59733  36-3J9 
1.61987  41-073 
1.64325  45-693 

0-5 
0-55 

0.6 
0.65 
0.7 

I.37I53  52-707 
1.39976  56.957 
1.42675  61.043 
1.45193  64.981 
1.47479  68.791 

1.43950   51.981 
1.46641    56.268 
1.49220   60.403 
1.51630   64.401 
1.53820   68.280 

1.51128  51.323 
I.53694  55-640 
1.56156  59.818 
1.58460  63.870 
1.60558  67.811 

1.58701   50.725 
1.61147  55.069 
1.63497  59.284 
1.65699  63.384 
1.67705  67.380 

1.66688  50.184 
1.69018  54-549 
1.71260  58.797 
I.73363  62.939 
1.75282  66.986 

0-75 
0.8 
0.85 
0.9 
0-95 

1.49487  72-491 
1.51182  76.101 
1.52532  79.638 
I-535I3  83.122 
1.54108  86.724 

1-55747   72.056 
J-57374  75-747 
1.58671   79-369 
1.59614  82.941 
1.60187  86.479 

1.62404   71.656 
1-63965   75-421 
1.65211   79.121 
1.66117  82.774 
1.66667  86.395 

1.69472   71.287 
1.70971   75.120 
1.72166   78.893 
1.73036  82.620 
1.73564  86.317 

I.76975  70-950 
1.78409  74.844 
1-79555  78-682 
1.80389  82.478 
1.80895  86.246 

1.05 
i.i 

1.2 

1.54308  90.000 
1.54108  93.276 
I-535I3  96-878 
1.52532  100.362 
1.51182  103.899 

1.60379  90.000 
1.60187  93-521 
1.59614  97-059 
1.58671  100.631 
1.57374  104.253 

1.66852  90.000 
1.66667  93-605 
1.66117  97.226 
1.65211  100.879 
1.63965  104.579 

1.73741  90.000 
I.73564  93-683 
1.73036  97.380 
1.72166  101.107 
1.70971  104.880 

i.  81066  90.000 
1.80895  93-754 
1.80389  97-522 
1-79555  101.318 
1.78409  105.156 

1.25 
1.4 

i-45 

1.49487  107.509 
1.47479  111.209 
1.45193  115.019 
1.42675  118.957 
1.39976  123.043 

1.55747  107.944 
1.53820  111.720 
1.51630  115.599 
1.49220  119.597 
1.46641  123.732 

1.62404  108.344 
1.60558  112.189 
1.58460  116.130 
1.56156  120.182 
1.53694  124.360 

1.69472  108.713 
1.67705  112.620 
1.65699  116.616 
1.63497  120.716 
1.61147  124.931 

1.76975  109.050 
1.75282  113.014 
1-73363  117-061 
1.71260  121.203 
1.69018  125.451 

i-SS 
1.6 
1.65 

1.37153  127.293 
1.34272  131.724 
1.31400  136.349 
1.28612  141.179 
1.25984  146.216 

1.43950  128.019 
1.41207  132.470 
1.38479  137.098 
1.35837  141.910 
1  -3335  1  146.907 

1.51128  128.677 
1.48517  133.145 
1.45926  137.773 
1.43421  142.565 
1.41070  147.523 

1.58701  129.275 
1.56217  133-755 
I.53756  138.380 
1.51381  143-153 
1.49155  148.074 

1.66688  129.816 
1.64325  134-307 
1.61987  138.927 
1-59733  143-681 
1.57626  148.567 

1-75 
1.8 
1.85 
1.9 

1.23594  151.459 
1.21515  156.895 
1.19816  162.504 
1.18552  168.252 
1.17782  174.100 

1.31095  152.085 
1.29137  157.432 
1.27540  162.929 
1.26358  168.547 
1.25631  174-252 

1.38939  152.641 

I-37093  I57-908 
1.35590  163.306 
1.34478  168.808 
1-33795  174-385 

1.47141  153.136 
1.45399  158.330 
1.47983  163.639 
1.41936  169.039 
1.42294  174-503 

1.55721  153.579 
1.54077  158.706 
1.52741  163.935 
1.51755  169.243 
1.51150  174.607 

2.00 

1.17520  180.000 

1.25386  180.000 

I-33565  180.000 

1.42078  180.000 

1.50946  180.000 

Example. 

sinh  (1.20  +  i  1.25) 

=  1.76975^109^.050 

=  1.76995  /i090.03'.oo*. 

[94] 


TABLE  X.    HYPERBOLIC   SINES,     sinh  (x  +  iq)  =  r /y.     CONTINUED 


=  1.25 


x  =  1.3 


*  =  i-3S 


x  =  1.4 


x  =  1.45 


o 
0.05 

O.I 

0.15 

O.2 

1.60192  o.ooo 
1.60384  5.301 
1.60954  10.576 
1.61884  15.803 
1.63145  20.958 

1.69838   o.ooo 
1.70019   5.218 
1.70557   10.415 
1.71435   15.568 
1.72627   20.659 

1.79909   o.ooo 
1.80080   5.145 
1.80588  10.271 
1.81417   15-359 
1.82544  20.392 

1.90430   o.ooo 
1.90592   5.080 
1.91072   10.143 
1.91856   15.172 
1.92921   20.153 

2.01427  o.ooo 

2.OI58O  5-021 
2.02034  IO.028 
2.02775  15.005 
2.03784  19-939 

0.25 

o-35 
0.4 

0-45 

1.64699  26.026 

1.66501  30.991 
1.68497  35.845 
1.70635  40.580 
1.72856  45-195 

1.74096   25.673 
1.75802  30.595 
1.77694  35.418 
1.79722  40.135 
1.81832   44-745 

1.83934  25.356 
I.85549  30.240 
I.87343  35-035 
1.89268  39.735 
1.91273  44-338 

1.94237   25.073 
1.95767   29.921 
1.97468  34.689 
1.99295   39.373 

2.OI2OO   43.97O 

2.O5O3O  24.8l8 
2.06479  29.634 
2.08093  34-379 
2.09828  39.O47 
2.II637  43.638 

o-S 
0-55 
0.6 
0.65 
0.7 

1.75104  49-693 
1.77323  54-077 
1.79462  58.354 
1.81470  62.533 
1.83304  66.625 

1.83970  49.248 
1.86084   53.648 
1.88123   57.950 
1.90040  62.163 
1.91792   66.295 

1.93306  48.845 
i-953I9  53-258 
1.97262  57-583 
1.99091  61.825 
2.00764  65.994 

2.03135   48.480 
2.05051   52.905 
2.06903   57-249 
2.08647   61.518 
2.10244   65.719 

2.13478  48.150 
2.15302  52.584 
2.17067  56.945 
2.I873I  61.238 
2.20254  65.469 

0-75 
0.8 
0.85 
0.9 
o-95 

1.84924  70.640 

1.86297  74.590 
1.87394  78.489 
1.88193  82.348 
1.88679  86.180 

1.93341   70.356 
1.94654   74-358 
1.95704   78.311 
1.96470  82.228 
1.96935  86.120 

2.02245   70.097 
2.03500  74-145 
2.04505   78.149 
2.05238  82.118 
2.05684  86.065 

2.11658   69.861 
2.12858   73.951 
2.13819   78.00O 
2.I452O   82.0l8 
2.14947   86.014 

2.2I6O4  69.645 

2-22751  73-774 
2.23669  77.864 
2.24334  81.925 
2.24747  85.968 

I.O 

1.05 
i.i 
i.i5 

1.2 

1.88842  90.000 
1.88679  93.820 
1.88193  97.652 
1.87394  101.511 
1.86297  105.410 

1.97091   90.000 
1.96935  93.880 
1.96470  97-772 
1.95704  101.689 
1.94654  105.642 

2.05833  90.000 
2.05684  93-935 
2.05238  97.882 
2.04505  101.851 
2.03500  105.855 

2.15090   9O.OOO 
2.14947   93.986 
2.14520   97-982 
2.13819  I02.OOO 
2.12858  IO6.O49 

2.24884  90.000 
2.24747  94-032 
2.24334  98-075 
2.23669  102.136 
2.22751  106.226 

1.25 

i-35 
1.4 

1-45 

1.84924  109.360 

1.83304  113.375 

1.81470  117.467 
1.79462  121.646 

1.77323  125.923 

1.93341  109.644 
1.91792  113.705 
1.90040  117.837 
1.88123  122.050 
1.86084  126.352 

2.02245  109.903 
2.00764  114.006 
1.99091  118.175 
1.97262  122.417 
1.95319  126.742 

2.11658  IIO.I39 
2.IO244  II4.28I 
2.08647  118.482 
2.06903  122.751 
2.05051  127.095 

2.21604  110.355 
2.20254  H4-531 
2.18731  118.762 
2.17067  123.055 
2.15302  127.416 

1-55 
1.6 

1.65 

1.75104  130.308 
1.72856  134.805 
1.70635  139.420 
1.68497  144.155 

1.66501  149.009 

1.83970  130.752 
1.81832  135.255 
1.79722  139.865 
1.77694  144.582 
1.75802  149-405 

1.93306  131.155 
1.91273  135.662 
1.89268  140.265 
1.87343  144-965 
1.85549  149.760 

2.03135  131.520 
2.OI20O  136.030 
1.99295  140.627 
1.97468  145-3" 
L95767  150.079 

2.13478  131.851 
2.11637  136.362 
2.09828  140.953 
2.08093  I45-62I 
2.06479  150.366 

1-75 
1.8 

1.85 
1.9 
1-95 

1.64699  153.974 
1.63145  159.042 

1.61884  164.197 
1.60954  169.424 
1.60384  174.699 

1.74096  154-327 
1.72627  159-341 
1.71435  164.432 
1.70557  169.585 
1.70019  174.782 

1.83934  154-644 
1.82544  159.608 
1.81417  164.641 
1.80588  169.729 
1.80080  174.855 

1.94237  I54.927 
I.9292I  159.847 
1.91856  164.828 
1.91072  169.857 
1.90592  174.920 

2.05030  155.182 
2.03784  160.061 
2.02775  164.995 
2.02034  169.972 
2.01580  174.979 

2.O 

1.60192  180.000 

1.69838  180.000 

1.79909  180.000 

1.90430  iSo.OOO 

2.01427  180.000 

Example. 

sinh  (1.45  +  i  1.70) 

=  2.06479  /I50°.366 

=  2.06479  /I  SO0.  2,2 

'.01'. 

[95] 


TABLE  X.    HYPERBOLIC   SINES,     sinh  (*  +  iq)  =  r  /V     CONTINUED 


x  =  1.50 


X  =    1 .60 


x  =  1.65 


a;  =  1.70 


o 

2.12928 

o.ooo 

2.24961 

o.ooo 

2-37557 

o.ooo 

2.50746 

o.ooo 

2.64563 

O.OOO 

0.05 

2.13073 

4.969 

2.25098 

4-923 

2.37686 

4.881 

2.50869 

4.843 

2.64679 

4.809 

O.I 

2.13502 

9-925 

2.25504 

9.833 

2.38071 

9-751 

2.51234 

9.677 

2.65025 

9.610 

0.15 

2.14204 

14-855 

2.26169 

14.721 

2.38701 

14.600 

2.51831 

14.492 

2.65591 

14-395 

0.2 

2-15159 

19.746 

2.27074 

19-574 

2.39558 

19.419 

2.52644 

19.280 

2.66361 

19.155 

0.25 

2.16340 

24.590 

2.28193 

24-385 

2.40619 

24.200 

2.53650 

24.034 

2.67316 

23.884 

°-3 

2.17714 

29.376 

2.29496 

29.144 

2.41856 

28.935 

2.54824 

28.747 

2.68430 

28.578 

0-35 

2.19245 

34.099 

2.30949 

33-846 

2-43235 

33-6l9 

2.56133 

33-415 

2.69673 

33.229 

0.4 

2.20892 

38.753 

2.32513 

38.488 

2-44721 

38.248 

2-57544 

38.032 

2.71014 

37.837 

0-45 

2.22612 

43-337 

2.34148 

43.066 

2.46274 

42.820 

2.59021 

42.599 

2.72418 

42.398 

0-5 

2.24362 

47.850 

2.35812 

47.580 

2-47857 

47-334 

2.60527 

47.113 

2.73850 

46.911 

o-SS 

2.26098 

52-294 

2.37465 

52.030 

2.49430 

Si-791 

2.62024 

51.574 

2.75274 

51-378 

0.6 

2.27779 

56.670 

2.39066 

56-420 

2-50955 

56.192 

2.63476 

55.986 

2.76657 

55-799 

0.65 

2.29365 

60.984 

2.40577 

60.752 

2-52395 

60.542 

2.64847 

60.351 

2.77963 

60.178 

0.7 

2.30819 

65.241 

2.41976 

65-033 

2.53717 

64.845 

2.66108 

64.673 

2.79164 

64-517 

0-75 

2.32107 

69.448 

2.43*93 

69.268 

2.54890 

69.105 

2.67226 

68.956 

2.80230 

68.821 

0.8 

2.33202 

73.611 

2.44238 

73-464 

2-55887 

73-329 

2.68178 

73.206 

2.81138 

73-094 

0.85 

2.34080 

77.740 

2.45076 

77.626 

2.56687 

77-523 

2.68941 

77.429 

2.81867 

77-343 

0.9 

2.34720 

81.842 

2.45688 

81.765 

2.57272 

81.695 

2.69499 

81.631 

2.82399 

Si-573 

o-95 

2.35110 

85-925 

2.46061 

85-887 

2.57627 

85-851 

2.69839 

85.819 

2.82723 

85.790 

I.O 

2.35241 

90.000 

2.46186 

90.000 

2.57746 

90.000 

2.69951 

90.000 

2.82832 

90.000 

1.05 

2.35110 

94-075 

2.46061 

94-113 

2.57627 

94.149 

2.69839 

94.181 

2.82723 

94.210 

i.i 

2.34720 

98.158 

2.45688 

98.235 

2.57272 

98-305 

2.69499 

98.369 

2.82399 

98.427 

i-iS 

2.34080 

102.260 

2.45076 

102.374 

2.56687 

102.477 

2.68941 

102.571 

2.81867 

102.657 

1.2 

2.33202 

106.389 

2.44238 

106.536 

2.55887 

106.671 

2.68178 

106.794 

2.81138 

106.906 

1-25 

2.32107 

110.552 

2.43*93 

110.732 

2.54890 

110.895 

2.67226 

111.044 

2.80230 

111.179 

i-3 

2.30819 

114-759 

2.41976 

114.967 

2.53717 

115-155 

2.66108 

"5-327 

2.79164 

115-483 

i-3S 

2.29365 

119.016 

2.40577 

119.248 

2-52395 

119.458 

2.64847 

.119.649 

2.77963 

119.822 

1.4 

2.27779 

123.330 

2.39066 

123.580 

2-50955 

123.808 

2.63476 

124.014 

2.76657 

124.201 

1-45 

2.26098 

127.706 

2.37465 

127.970 

2.49430 

128.209 

2.62024 

128.426 

2.75274 

128.622 

i-5 

2.24362 

132.150 

2.35812 

132.421 

2.47857 

132.666 

2.60527 

132.887 

2.73850 

133-089 

i-55 

2.22612 

136.663 

2.34148 

136.934 

2.46274 

137.180 

2.59021 

137.401 

2.72418 

137.602 

1.6 

2.20892 

141.247 

2.32513 

141.512 

2.44721 

141.752 

2-57544 

141.968 

2.71014 

142.163 

1.65 

2.19245 

145.901 

2.30949 

146.154 

2-43235 

146.381 

2-56133 

146.585 

2.69673 

146.771 

i-7 

2.17714 

150.624 

2.29496 

150.856 

2.41856 

151.065 

2.54824 

I5L253 

2.68430 

151.422 

i-7S 

2.16340 

I55.4IO 

2.28193 

I55-6I5 

2.40619 

155.800 

2.53650 

155.966 

2.67316 

156.116 

1.8 

2-15159 

160.254 

2.27074 

160.426 

2.39558 

160.581 

2.52644 

160.720 

2.66361 

160.845 

1.85 

2.14204 

165.145 

2.26169 

165.279 

2.38701 

165.400 

2.51831 

165.508 

2.65591 

165.605 

1.9 

2.13502 

170.075 

2.25504 

170.167 

2.38071 

170.249 

2.51234 

170.323 

2.65025 

170.390 

i-95 

2.13073 

I75-03I 

2.25098 

175.077 

2.37686 

175-119 

2.50869 

I75-I57 

2.64679 

I75-I9I 

2.O 

2.12928 

180.000 

2.24961 

180.000 

2-37557 

180.000 

2.50746 

180.000 

2.64563 

180.000 

Example. 

sinh  (1.55 

+  i  0.60) 

=  2.39066  /56°.42O 

=  2.39066  /56°.  25'. 

12*. 

[96] 


TABLE  X.     HYPERBOLIC   SINES,     sinh  (*  +  iq)  =  r  fy.     CONTINUED 


x  = 

i-75 

*  = 

1.8 

x  = 

1.85 

x  = 

1.9 

x  = 

i-95 

9 

r 

7 

r 

V 

r 

T 

r 

•y 

r 

7 

o 

o 

o 

0 

0 

0 

2.79041 

o.oo 

2.94217 

o.oo 

3 

.10129 

o.oo 

3.26816 

0.00 

3-44321 

o.oo 

0.05 

2.79151 

4-779 

2.94322 

4.752 

3 

.10228 

4.727 

3.26911 

4-705 

3.44410 

4.685 

O.I 

2.79479 

9-551 

2-94633 

9-497 

3 

10523 

9.448 

3.27191 

9-405 

3-44675 

9-365 

0.15 

2.80016 

I4-307 

2.95142 

14.228 

3 

,11006 

14.158 

3.27649 

14.094 

3-45H2 

14.036 

0.2 

2.80747 

19.042 

2-95835 

18.941 

3 

11665 

18.850 

3.28274 

!8.767 

3-45702 

18.693 

0.25 

2.81653 

23-750 

2.96695 

23.629 

3' 

12481 

23.520 

3.29049 

23.421 

3.46441 

23-332 

o-3 

2.82710 

28.425 

2-97699 

28.287 

3' 

!3434 

28.163 

3-29955 

28.051 

3-47301 

27.950 

o-3S 

2.83891 

33-063 

2.98821 

32.912 

3- 

14500 

32.776 

3.30967 

32-654 

3.48263 

32.543 

0.4 

2.85165 

37.661 

3.00031 

37-501 

3- 

15650 

37-357 

3.32060 

37-227 

3-49302 

37.110 

0-45 

2.86499 

42.216 

3.01300 

42-053 

3- 

16856 

41.904 

3-33207 

41.770 

3-50393 

41.649 

o-5 

2.87861 

46.730 

3-02595 

46-565 

3- 

18088 

46.416 

3-34378 

46.281 

3-5J507 

46.160 

o-SS 

2.89217 

51.200 

3-03885 

5I-039 

3- 

I93I5 

50-893 

3-35546 

50.761 

3.52617 

50.642 

0.6 

2.90532 

55-630 

3-05138 

55-4/6 

3' 

20507 

55-337 

3.36681 

55-211 

3-53698 

55.096 

0.65 

2.91777 

60.020 

3.06323 

59-877 

3- 

21636 

59-748 

3-37756 

59-63I 

3-54721 

59.524 

0.7 

2.92921 

64-375 

3-07413 

64.246 

3- 

22675 

64.1  29 

3-38745 

64.023 

3-55663 

63.927 

0-75 

2-93938 

68.698 

3.08382 

68.586 

3' 

23598 

68.484 

3-39624 

68.392 

3-56500 

68.308 

0.8 

2.94804 

72.992 

3.09206 

72.900 

3- 

24384 

72.816 

3-40373 

72.740 

3-572I3 

72.670 

0.85 

2.95498 

77.265 

3-09869 

77-194 

3- 

25015 

77.129 

3-40975 

77-071 

3-57788 

77.017 

0.9 

2.96006 

81.520 

3-J0353 

81.471 

3- 

25477 

81.428 

3-4i4i5 

81.388 

3.58207 

81.352 

0-95 

2.96315 

85-763 

3.10648 

85-738 

3- 

25759 

85-716 

3.41683 

85.696 

3.58462 

85.678 

I.O 

2.96419 

90.000 

3.10747 

90.000 

3- 

25853 

90.000 

34I773 

90.000 

3-58548 

90.000 

1.05 

2.96315 

94-237 

3.10648 

94.262 

3' 

25759 

94.284 

3.41683 

94-304 

3.58462 

94.322 

i.i 

2.96006 

98.480 

3-10353 

98.529 

3. 

25477 

98.572 

3-41415 

98.612 

3.58207 

98.648 

i-i5 

2.95498 

102.735 

3.09869 

102.806 

3 

25015 

102.871 

340975 

102.929 

3-57788 

102.983 

1.2 

2.94804 

107.008 

3.09206 

107.100 

3 

24384 

107.184 

3.40373 

107.260 

3-572I3 

107.330 

I-2S 

2-93938 

111.302 

3.08382 

111.414 

3 

•23598 

111.516 

3-39624 

111.608 

3-56500 

111.692 

i-3 

2.92921 

115-625 

3-074I3 

115-754 

3 

.22675 

115.871 

3-38745 

"5-977 

3-55663 

116.073 

i-3S 

2.91777 

119.980 

3-06323 

120.123 

3 

.21636 

120.252 

3-37756 

120.369 

3-54721 

120.476 

i-4 

2.90532 

124.370 

3-05138 

124.524 

3 

.20507 

124.663 

3.36681 

124.789 

3-53698 

124.904 

i  -45 

2.89217 

128.800 

3-03885 

128.961 

3 

•I93I5 

129.107 

3-35546 

129.239 

3.52617 

129.358 

i-5 

2.87861 

133.270 

3-02595 

133-435 

3 

.18088 

133-584 

3-34378 

I33-7I9 

3-5I507 

133.840 

i-SS 

2.86499 

I37-784 

3.01300 

137-947 

3 

.16856 

138.096 

3-33207 

138.230 

3-50393 

138.351 

1.6 

2.85165 

142.339 

3.00031 

142.499 

3 

•15650 

142.643 

3.32060 

142.773 

3-49302 

142.890 

1.65 

2.83891 

146.937 

2.98821 

147.088 

3 

.14500 

147.224 

3-30967 

147.346 

3.48263 

147-457 

i-7 

2.82710 

I5I-575 

2-97699 

I5L7I3 

3 

•13434 

151-837 

3-29955 

151.949 

3-47301 

152.050 

i-7S 

2.81653 

156.250 

2-96695 

156.371 

3 

.12481 

156.480 

3.29049 

156.579 

3.46441 

156.668 

1.8 

2.80747 

160.958 

2-95835 

161.059 

3 

.11665 

161.150 

3.28274 

161.233 

3-45702 

161.307 

1.85 

2.80016 

165.693 

2.95142 

165.772 

3 

.11006 

165.842 

3-27649 

165.906 

3-45ii2 

165.964 

1.9 

2.79479 

170.449 

2-94633 

170.503 

3 

•10523 

170.552 

3.27191 

170.595 

3-44675 

170.635 

1-95 

2.79151 

175.221 

2.94322 

175.248 

3 

.10228 

I75.273 

3.26911 

I75.295 

3.44410 

J75-3I5 

2.O  2.79041      iSo.OOO         2.94217     iSo.OOO         3.IOI29     iSo.OOO         3.26816     iSo.OOO         3.44321      iSo.OOO 

Example.       sinh  (1.90  +  *  2.0)  =  3.26816  /i8o°.o  =  3.26816  \i8o°.o. 


[97] 


TABLE  X.    HYPERBOLIC  SINES,     sinh  (x  +  iq)  =  r /y.     CONTINUED 


X  =  2.O 

*  =  2.05 

*  =  2.1 

*  =  2.15 

*  =  2.2 

g 

r     i 

f      7 

r     7 

r     V 

f      V 

o 

o 

o 

o 

o 

0 

3.62686   o.ooo 

2.81950   o.ooo 

4.02186   o.ooo 

4.23419   o.ooo 

4.45711   o.ooo 

0.05 

3.62771  4-667 

3.82039   4.651 

4.02263   4.636 

4.23491   4.623 

4.45779   4.611 

O.I 

3.63023  9.330 

3.82279   9.298 

4.02490   9.269 

4.23707  9.243 

4.45985   9.220 

0.15 

3.63437  13.984 

3.82671  13.938 

4.02863  13.895 

4.24061  13.857 

4.46322  13.823 

0.2 

3.64000  18.626 

3.83206  18.566 

4.03371  18.511 

4.24544  18.462 

4.46780  18.418 

0.25 

3.64699  23.252 

3.83870  23.180 

4.04002   23.114 

4.25145  23.055 

4.47350  23.002 

o-3 

3.65517  27.858 

3.84648  27.776 

4.04740  27.701 

4.25846  27.634 

4.48017  27.573 

0-35 

3.66430  32.443 

3.85515  32.353 

4.05566  32.271 

4.26630  32.197 

4.48763  32.130 

0.4 

3.67418  37.004 

3.86454  36.908 

4.06459  36.821 

4-27479  36.743 

4.49570  36.671 

0-45 

3.68455  41.539 

3.87440  41.440 

4.07396  41-351 

4.28371  41.270 

4.50417  41.196 

0-5 

3.69515  46.049 

3.88448  45-950 

4.08355  45.859 

4.29282  45.778 

4.51285  45.703 

o-SS 

3.70572  50.533 

3-89453  50-435 

4.09311  50.347 

4-3OI93   50-266 

4.52150  50.193 

0.6 

3.71600  54-992 

3.90432  54.898 

4.10243  54.813 

4.31079   54.736 

4-52993  54-666 

0.65 

3.72574  59-427 

3-91359  59-340 

4.11125  59.260 

4.31918   59.188 

4-53793  59-123 

0.7 

3-73470  63.840 

3.92213  63.761 

4.11938  63.689 

4.32693   63.624 

4.54529  63.565 

0-75 

3.74268  68.232 

3.92972  68.164 

4.12661  68.101 

4.33381   68.044 

4.55184  67.993 

0.8 

3.74948  72.608 

3.93620  72.551 

4.13278   72-499 

4.33968   72-452 

4-55744  72.409 

0.85 

•3-75495   76-969 

3.94142  76.925 

4.13774  76.885 

4.34440   76.849 

4.56167   76.816 

0.9 

3.75894  81.319 

3.94521  81.289 

4.14136  81.262 

4.34785   81.237 

4.56523  81.215 

o.QS 

3.76137  85.661 

3-94753  85.646 

4.14357  85.632 

4.34996  85.620 

4-56723  85.609 

1.0 

3.76220  oo.ooo 

3.94832  90.000 

4.14431  90.000 

4-35o67   go-000 

4.56791   90.000 

1.05 

3.76137  94.339 

3-94753  94-354 

4-14357  94-368 

4.34996  94-38o 

4.56723  94.391 

.1 

3.75894  98.681 

3.94521  98.711 

4.14136  98-738 

4.34785  98.763 

4-56523  98-785 

•IS 

3.75495  103.031 

3.94142  103.075 

4.13774  103.115 

4.34440  103.151 

4.56167  103.184 

.2 

3.74948  107.392 

3.93620  107.449 

4.13278  107.501 

4.33968  107.548 

4-55744  107.59! 

•25 

3.74268  111.768 

3.92972  111.836 

4.12661  111.899 

4.33381  111.956 

4.55184  112.007 

•3 

3.73470  116.160 

3.92213  116.239 

4.11938  116.311 

4.32693  116.376 

4.54529  116.435 

•35 

3.72574  120.573 

3.91359  120.660 

4.11125  120.740 

4.31918  120.812 

4-53793  120.877 

•4 

3.71600  125.008 

3.90432  125.102 

4.10243  125.187 

4.31079  125.264 

4.52993  125.334 

145 

3.70572  129.467 

3.89453  129.565 

4.09311  129.653 

4.30193  129.734 

4.52150  129.807 

i-5 

3.69515  133.951 

3.88448  134-051 

4-08355  134-141 

4.29282  134.223 

4.51285  134-297 

i-SS 

3.68455  138.461 

3.87440  138.560 

4.07396  138.649 

4.28371  138.730 

4.50417  138.804 

1.6 

3.67418  142.996 

3.86454  143.092 

4.06459  143-179 

4.27479  143-257 

4.49570  143-329 

1.65 

3.66430  147-557 

3.85515  147-647 

4.05566  147-729 

4.26630  147.803 

4.48763  147.870 

i-7 

3.65517  152.142 

3.84648  152.224 

4.04740  152.299 

4.25846  152.366 

4.48017  152.427 

•75 

3.64699  156.748 

3.83870  156.820 

4.04002  156.886 

4.25145  156.945 

4.47350  156.998 

.8 

3.64000  161.374 

3.83206  161.434 

4.03371  161.489 

4.24544  161.538 

4.46780  161.582 

•85 

3.63437  166.016 

3.82671  166.062 

4.02863  166.105 

4.24061  166.143 

4.46322  166.177 

•9 

3.63023  170.670 

3.82279  170.702 

4.02490  170.731 

4.23707  170.757 

445985  170.780 

•95 

3.62771  175.333 

3.82039  175-349 

4.02263  175.364 

4.23491  175-377 

4-45779  175-389 

2.0 

3.62686  180.000 

3.81958  180.000 

4.02186  180.000 

4.23419  180.000 

4.45711  180.000 

Example.^    sinh  (2.0  +  i  i.o)  =  3.76220  /go0. 


[98] 


TABLE  X.    HYPERBOLIC  SINES,     sinh  (x  +  ig)  =  r  fy.    CONTINUED 


*  =  2.25 


x  =  2.3 


*  =  2.35 


x  =  2.4 


x  =  245 


0 

4.69117 

o.ooo 

4.93696 

o.ooo 

5 

19510 

o.ooo 

5.46623 

o.ooo 

5-75*03 

o.ooo 

0.05 

4.69182 

4.601 

4-93759 

4-591 

5 

19569 

4.582 

5.46679 

4-574 

5.75I56 

4-567 

O.I 

4-69377 

9.199 

4-93944 

9.180 

5 

I974S 

9.163 

5.46847 

9.147 

5-753i6 

9-133 

0.15 

4.69697 

13-792 

4.94249 

13-764 

5- 

20035 

13.739 

5.47122 

13.716 

5-75576 

13-695 

0.2 

4.70134 

18.378 

4.94662 

18.341 

5 

20428 

18.309 

5-47495 

18.279 

5-75932 

18.252 

0.2S 

4.70675 

22.954 

4-95I77 

22.910 

5 

20917 

22.871 

5.47961 

22.835 

5-76375 

22.803 

o-3 

4.71308 

27.518 

4.95780 

27.469 

•5 

21490 

27.424 

5-48505 

27-383 

5-76892 

27-347 

0-35 

4.72017 

32.070 

4.96454 

32.016 

5 

22131 

31.966 

549II5 

31.922 

5.77472 

31.881 

0.4 

4.72784 

36.608 

4.97183 

36.549 

5 

22825 

36-497 

549775 

36449 

5.78099 

36.407 

°4S 

4-73592 

41.130 

4-97950 

41.070 

5 

23553 

41.016 

5.50468 

40.966 

5.78758 

40.922 

°S 

4.74415 

45-636 

4-98734 

45-576 

5 

23728 

45.521 

5.5"77 

45471 

5-79434 

45427 

o-SS 

4.75240 

50.127 

4.99518 

50.068 

5 

25046 

50.014 

5-51887 

49.965 

5.80108 

49.921 

0.6 

4.76042 

54-603 

5.00281 

54-546 

5 

25772 

54-494 

5.52578 

54-447 

5.80765 

54405 

0.65 

4.76802 

59.064 

5.01005 

59-on 

5 

26461 

58.962 

5.53233 

58.919 

5-81389 

58.879 

0.7 

4-77503 

63.512 

5.01672 

63-463 

5 

27096 

63.419 

5.53837 

63.380 

5.81964 

63-344 

0-75 

4.78127 

67.947 

5.02265 

67.904 

5 

27662 

67.866 

5-54375 

67-831 

5.82476 

67.800 

0.8 

4.78661 

72.371 

5-02773 

72.336 

5 

28143 

72.304 

5-54834 

72.275 

5-82913 

72.249 

0.85 

4.79088 

76.786 

5.03181 

76.759 

5 

28531 

76.735 

5-55204 

76.712 

5-83265 

76.692 

0.9 

4.79402 

81.195 

5-03479 

81.176 

5 

28816 

81.160 

5-55474 

81.144 

5-83522 

81.131 

o-95 

4-79593 

85-599 

5.03661 

85-589 

5 

28989 

85.581 

5.55639 

85-573 

5-83680 

85.566 

1.0 

4-79657 

90.000 

5.03722 

90.000 

5 

29047 

90.000 

5.55695 

90.000 

5.83732 

90.000 

1.05 

4-79593 

94.401 

5.03661 

94.411 

5 

28989 

94.419 

5.55639 

94.427 

5.83680 

94434 

i.i 

4.79402 

98.805 

5-03479 

98.824 

5 

28816 

98.840 

5-55474 

98.856 

5-83522 

98.869 

i-iS 

4.79088 

103.214 

5.03181 

103.241 

5 

28531 

103.265 

5-55204 

103.288 

5-83265 

103.308 

1.2 

4.78661 

107.629 

5-02773 

107.664 

5 

28143 

107.696 

5.54834 

107.725 

5-82913 

107.751 

I-2S 

4.78127 

112.053 

5.02265 

112.096 

5 

27662 

112.134 

5-54375 

112.169 

5.82476 

112.  2OO 

i-3 

4-77503 

116.488 

5.01672 

"6.537 

5 

27096 

116.581 

5-53837 

116.620 

5.81964 

116.656 

i-35 

4.76802 

120.936 

5.01005 

120.989 

5 

26461 

121.038 

5-53233 

121.081 

5-81389 

121.  121 

14 

4.76042 

125-397 

5.00281 

125-454 

5 

25772 

125.506 

5.52578 

125.553 

5-80765 

125.595 

145 

4.75240 

129.873 

4.99518 

129.932 

5 

25046 

129.986 

5-51887 

130-035 

5.80108 

130.079 

-5 

4-744I5 

134-364 

4-98734 

134.424 

5 

23728 

134-479 

5-5"77 

134.529 

5-79434 

134-573 

•55 

4-73592 

138.870 

4-9795° 

138.930 

5 

23553 

138.984 

5.50468 

I39.034 

5-78758 

139.078 

.6 

4.72784 

143-392 

4.97183 

I4345I 

5 

22825 

143-503 

549775 

I43.55I 

5.78099 

143-593 

.65 

4.72017 

147.930 

4-96454 

147.984 

S 

22131 

148.034 

549"5 

148.078 

5-77472 

I48.II9 

•7 

4.71308 

152.482 

4.95780 

152-531 

5 

21490 

152.576 

548505 

152.617 

5-76982 

152.653 

•75 

4-70675 

157.046 

4.95I77 

157.090 

5 

20917 

157.129 

5.47961 

157-165 

5.76375 

I57-I97 

.8 

4-70I34 

161.622 

4.94662 

161.659 

5 

20428 

161.691 

547495 

161.721 

5-75932 

161.748 

•85 

4.69697 

166.208 

4.94249 

166.236 

5 

20035 

166.261 

5.47122 

166.284 

5.75576 

166.305 

•9 

4.69377 

170.801 

4-93944 

170.820 

5 

I974S 

170.837 

5.46847 

170.853 

5-753i6 

170.867 

1-95 

4.69182 

175.399 

4-93759 

175.409 

5 

19569 

175.418 

5.46679 

175.426 

5-75*56 

175433 

2.O 

4.69117 

180.000 

4.93696 

180.000 

5 

19510 

180.000 

546623 

180.000 

5-75I03 

iSo.OOO 

Example.      sinh  (2.40  +  *  0.4)  =  549775  /36°-449  =  5-49775  /36°. 26^.56'. 


[99] 


TABLE  X.    HYPERBOLIC  SINES,     sinh  (x  +  iq)  =  r /y.    CONTINUED 


x  =  2.5 


=  2.55 


*  =  2.6 


x  =  2.65 


x  =  2.7 


o 
0.05 

O.I 

0.15 

O.2 

6.05020 
6.05071 
6.05223 
6.05471 
6.05808 

o.ooo 

4.561 

9.120 

13.676 
18.228 

6.36451 
6.36499 
6.36644 
6.36879 
6.37201 

o.ooo 

4-555 
9.109 
13.660 
18.206 

6.69473 
6.69518 
6.69656 
6.69880 
6.70185 

o.ooo 

4-550 

9.098 

13.644 
18.187 

7.04169 
7.04213 
7-04343 
7-04556 
7.04847 

0.000 

4-545 
9.089 

13-631 
18.169 

7.40626 
7.40668 
7.40791 
7.40994 
7.41271 

o.ooo 
4-540 
9.080 
13.618 
18.153 

O.25 

o-3 
0-35 
0.4 

0-45 

6.06229 
6.06722 
6.07273 
6.07869 
6.08496 

22.774 
27-314 

3!-845 
36.368 
40.882 

6.37601 
6.38068 
6.38592 
6.39160 

6-39755 

22.748 
27.284 
31.812 

36.333 
40.845 

6.70565 
6.71011 
6.71509 
6.72048 
6.72616 

22.725 
27.257 
31.782 
36.301 
40.813 

7.05208 
7.05631 
7.06105 
7.06618 
7.07158 

22.703 
27.232 
3I-756 
36.273 
40.783 

7.41614 

7.42017 
7-42467 
742955 
7.43468 

22.684 
27.210 

36.246 
40.756 

o-S 
0-55 
0.6 
0.65 
0.7 

6.09139 
6.09781 
6.10406 
6.10999 
6.11547 

45-386 
49.881 
54-366 
58-843 
63.3" 

6.40367 
6.40977 
6.41572 

6.42137 
6.42658 

45-350 
49.845 
54-332 
58.810 
63.282 

6.73197 
6.73778 

6-74343 
6.74881 

6.75376 

45-316 

49.812 

54-300 
58.781 
63.255 

7.07711 
7.08264 
7.08802 

7-09313 
7.09784 

45.286 
49.782 
54.272 

58.754 
63.230 

7-43994 
7-445I9 
745032 
7-455I8 
7.45966 

45-259 
49-755 
54.246 
58.730 
63.209 

o-75 
0.8 
0.85 
0.9 
0-95 

6.12034 
6.12450 
6.12785 
6.13030 
6.13179 

67.772 
72.226 
76.674 
81.119 
85.560 

6.43121 
6.43518 
6.43836 
6.44069 
6.44212 

67.746 
72.204 
76.658 
81.107 
85-554 

6.75818 
6.76195 
6.76499 
6.76720 
6.76855 

67.722 

72.185 
76.643 

81.097 

85.549 

7.10204 
7.10563 
7.10851 
7.11062 
7.11191 

67.701 
72.167 
76.629 
81.088 
85-544 

7.46366 
7.46708 
7.46982 
7-47I83 
7.47306 

67.682 
72.152 
76.617 
81.080 
85-540 

I.OO 

1.05 

I.I 

1-15 

1.2 

6.13229 
6.13179 
6.13030 
6.12785 
6.12450 

90.000 
94.440 
98.881 
103.326 
107.774 

6.44259 
6.44212 
6.44069 
6.43836 
6.43518 

90.000 
94-446 
98-893 
103.342 
107.796 

6.76901 
6.76855 
6.76720 
6.76499 
6.76195 

90.000 

94.451 
98.903 

103.357 
107.815 

7.11234 
7.11191 
7.11062 
7.10851 
7.10563 

90.000 
94.456 
98.912 
103.371 
107.833 

7-47347 
7.47306 
7-47i83 
7.46982 
7.46708 

90.000 
94.460 
98.920 

103-383 
107.848 

1-25 

i-35 
1.4 

1-45 

6.12034 
6.11547 
6.10999 
6.10406 
6.09781 

112.228 
116.689 
121.157 
125.634 
130.119 

6.43121 
6.42658 
6.42137 
6.41572 
6.40977 

112.254 
116.718 
121.190 
125.668 
130.155 

6.75818 
6.75376 
6.74881 
6-74343 
6-73778 

112.278 

116.745 

121.219 
125.700 

130.188 

7.10204 
7.09784 

7-093I3 
7.08802 
7.08264 

112.299 
116.770 
121.246 
125.728 
130.218 

7.46366 
7.45966 

745032 
7-445I9 

112.318 
116.791 
121.270 

125.754 
130.245 

i-55 
1.6 

1-65 

6.09139 
6.08496 
6.07869 
6.07273 
6.06722 

134.614 
139.118 
143.632 

148.155 
152.686 

6.40367 

6-39755 
6.39160 
6.38592 
6.38068 

134-651 
i39-I55 
143.667 
148.188 
152.716 

6.73197 
6.72616 
6.72048 
6.71509 
6.71011 

134.684 
139.187 
143.699 
148.218 

7.07711 
7.07158 
7.06618 
7.06105 
7.05631 

I34-7I3 
139.217 

I43-727 
148.244 
152.768 

7-43994 
7.43468 

7-42955 
7.42467 
7.42017 

I34.74I 
139.244 

143-754 
148.269 
152.790 

i-75 
1.8 
1-85 
1.9 
1-95 

6.06229 
6.05808 
6.05471 
6.05223 
6.05071 

157.226 
161.772 
166.324 
170.880 
175-439 

6.37601 
6.37201 
6.36879 
6.36644 
6.36499 

157-252 
161.794 
166.340 
170.891 
175-445 

6.70565 
6.70185 
6.69880 
6.69656 
6.69518 

157-275 
161.813 
166.356 

170.902 

175-450 

7.05208 
7.04847 
7-04566 

7-04343 
7.04213 

157.297 
161.831 
166.369 
170.911 
175-455 

7.41614 
7.41271 
7.40994 
7.40791 
7.40668 

157.316 
161.847 
166.382 
170.920 
175.460 

2.0 

6.05020 

180.000 

6.36451 

180.000 

6.69473 

180.000 

7.04169 

180.000 

7.40626 

180.000 

Example. 

sinh  (2.6 

+  io.6) 

=  6.74343 

/54°-300  =  6.74343  /54°-i8'oo". 

[  100  ] 


TABLE  X.    HYPERBOLIC  SINES,     sinh  (x  +  iq)  =  r  /y.     CONTINUED 


x  =  2.75 


=  2.85 


*  =  2.9 


x  =  2.95 


0 

7.78935 

o.ooo 

8.19192 

o.ooo 

8.61497 

o.ooo 

9-05956 

o.ooo 

9.52681 

o.ooo 

o.os 

7.78975 

4-537 

8.19230 

4-533 

8.61532 

4.530 

9.05990 

4-527 

9-52713 

4.525 

O.I 

7.79092 

9-073 

8.19341 

9.066 

8.61639 

9.060 

9.06091 

9-054 

9-52809 

9.049 

0.15 

7.79285 

13.607 

8.19524 

I3-596 

8.61813 

13-587 

9.06257 

13.579 

9.52967 

I3.57I 

0.2 

7-79547 

18.138 

8.19774 

18.125 

8.62051 

18.113 

9.06483 

18.102 

9.53182 

18.093 

0.25 

7-79875 

22.666 

8.20085 

22.650 

8.62347 

22.636 

9.06764 

22.623 

9-53450 

22.611 

7.80257 

27.190 

8.20449 

27.172 

8.62691 

27-155 

9.07093 

27.141 

27.127 

0-35 

7.80685 

31.709 

8.20857 

31.689 

8.63080 

31.671 

9.07461 

31.655 

9-54"3 

31.640 

0.4 

7.81149 

36.223 

8.21298 

36.202 

8.63500 

36.183 

9.07861 

36.165 

9-54492 

36-150 

0-45 

7-81637 

40.731 

8.21762 

40.710 

8.63941 

40.690 

9.08281 

40.671 

9.54892 

40.655 

o-5 

7.82138 

45-234 

8.22238 

45.212 

8.64395 

45.192 

9.08711 

45.173 

9-55301 

45-157 

0.55 

7.82638 

49-731 

8.22713 

49.709 

8.64846 

49.689 

9.09142 

49.671 

9-557" 

49.655 

0.6 

7-83125 

54-222 

8.23177 

54-201 

8.65287 

54.182 

9.09561 

54.165 

9.56109 

54-149 

0.65 

7-83588 

58.708 

8.23617 

58.688 

8.65706 

58.670 

9.09959 

58.654 

9.56488 

58.640 

0.7 

7-84015 

63.189 

8.24024 

63.171 

8.66092 

63-155 

9.10327 

63.140 

9-56838 

63.127 

0-75 

7.84395 

67-665 

8.24385 

67.649 

8.66436 

67.636 

9.10655 

67.622 

9.57I50 

67.611 

0.8 

7.84720 

72.137 

8.24694 

72.124 

8.66731 

72.112 

9.10934 

72.102 

9.57416 

72.092 

0.85 

7.84980 

76.606 

8.24942 

76-596 

8.66967 

76.587 

9.11160 

76.579 

9-57630 

76.571 

0.9 

7.85172 

8l.O72 

8.25124 

81.065 

8.67140 

81.059 

9.11324 

81.053 

9-57787 

81.048 

0-95 

7.85288 

85-536 

8-25235 

85-533 

8.67246 

85-530 

9.11424 

85.527 

9.57882 

85-524 

I.O 

7-85328 

90.000 

8.25273 

90.000 

8.67281 

90.000 

9.11458 

90.000 

9-579I5 

90.000 

1.05 

7.85288 

94.464 

8-25235 

94467 

8.67246 

94-470 

9.11424 

94-473 

9-57882 

94.476 

i.i 

7.85172 

98.928 

8.25124 

98.935 

8.67140 

98.941 

9.11324 

98.947 

9-57787 

98.952 

1.15 

7.84980 

103.394 

8.24942 

103.404 

8.66967 

103.413 

9.11160 

103.421 

9-57630 

103.429 

1.2 

7.84720 

107.863 

8.24694 

107.876 

8.66731 

107.888 

9.10934 

107.898 

9.57416 

107.908 

1-25 

7.84395 

"2-335 

8.24385 

"2.351 

8.66436 

112.364 

9.10655 

112.378 

9-57I50 

112.389 

7.84015 

116.811 

8.24024 

116.829 

8.66092 

116.845 

9.10327 

116.860 

9.56838 

116.873 

1-35 

7-83588 

121.292 

8.23617 

121.312 

8.65706 

121.330 

9.09959 

121.346 

9.56488 

121.360 

7-83125 

125.778 

8.23177 

125.799 

8.65287 

125.818 

9.09561 

125-835 

9.56109 

125-851 

1-45 

7.82638 

130.269 

8.22713 

130.291 

8.64846 

130.311 

9.09142 

130.329 

9-557" 

I30.345 

i-5 

7.82138 

134.766 

8.22238 

134.788 

8.64395 

134.808 

9.08711 

134.827 

9-55301 

134-843 

1-55 

7.81637 

139.269 

8.21762 

139.290 

8.63941 

139.310 

9.08281 

139-329 

9.54892 

139-345 

1.6 

7.81149 

143-777 

8.21298 

143.798 

8.63500 

143.817 

9.07861 

143-835 

9.54492 

143.850 

1.65 

7.80685 

148.291 

8.20857 

148.311 

8.63080 

148.329 

9.07461 

148-345 

9-54"3 

148.360 

7-80257 

152.810 

8.20449 

152.828 

8.62691 

152.845 

9.07093 

152.859 

152-873 

i-75 

7-79875 

157-334 

8.20085 

I57-350 

8.62347 

157-364 

9.06764 

157-377 

9-53450 

I57-389 

1.8 

7-79547 

161.862 

8.19774 

161.875 

8.62051 

161.887 

9.06483 

161.898 

9.53182 

161.907 

1-85 

7.79285 

166.393 

8.19524 

166.404 

8.61813 

166.413 

9.06257 

166.421 

9-52967 

166.429 

1.9 

7.79092 

170.927 

8.19341 

170.934 

8.61639 

170.940 

9.06091 

1  70.946 

9.52809 

170.951 

1-95 

7-7897S 

175-463 

8.19230 

175467 

8.61532 

175.470 

9.05990 

175-473 

9-52713 

175-475 

2.0 

7-7893S 

180.000 

8.19192 

180.000 

8.61497 

180.000 

9-05956 

180.000 

9.52681 

180.000 

Example. 

sinh  (2.95 

+  i  1.95) 

=  9-52713 

/i75°.475 

i  =  9-527I3  /I7S°-28'.30*. 

[I0t] 


TABLE  X.    HYPERBOLIC  SINES,     sinh  (x  +  iq)  =  r  /y.    CONTINUED 


*  =  3-o 

*  =  3-05 

x  =  3.1 

*  =  3-15 

X  =  3.2 

9 

r      y 

r      y 

r      y 

f      y 

r     y 

o 
0.05 

O.I 

0.15 

O.2 

0 

10.01787   o.ooo 
10.01820   4.522 
10.01910   9.044 
10.02060  13.565 
10.02260  18.084 

0 

10.53399   o.ooo 

10.53430  4.520  - 
10.53520   9.040 
10.53660  13.559 

I0-538so  18.076 

o 
11.07645   o.ooo 
11.07670   4.518 
11.07750   9.036 
11.07890  13.553 
11.08080  18.068 

0 

11.64661   o.ooo 
11.64690   4.516 

11.64770   9.033 
11.64890  13.548 
11.65070  18.062 

0 

12.24588  o.ooo 
12.24610  4.515 

12.24690  9.030 

12.24810  13.543 

12.24980  18.056 

0.25 

0-35 
0.4 

0-45 

10.02520  22.601 
10.02820  27.115 

10.03150  31.627 
10.03510  36.135 
10.03890  40.640 

10.54090  22.591 
10.54380  27.104 
10.54690  31.615 
10.55040  36.122 
10.55400  40.627 

11.08310  22.583 
11.08580  27.094 
11.08880  31.604 
11.09200  36.111 
11.09540  40.615 

11.65290  22.575 
11.65540  27.085 
11.65830  31.594 

11.66140  36.100 
11.66470  40.604 

12.25180  22.568 

12.25430  27.077 
12.25700  31.585 

12.26000  36.091 

12.26310  40.594 

o-S 
0-55 
0.6 
0.65 
o-7 

10.04280  45.142 
10.04670  49.640 

10.05050  54.135 

10.05410  58.626 

10.05743  63.114 

10.55770  45-129 
10.56140  49.627 
10.56500  54-122 
10.56840  58.614 
10.57160  63.104 

11.09900  45.116 
11.10250  49.615 
11.10600  54.110 
11.10920  58.604 

II.II22O   63.094 

11.66810  45.105 
11.67140  49.604 
11.67470  54-100 
11.67780  58.594 
11.68060  63.085 

12.26630  45.095 

12.26950  49.594 
12.27260  54.090 
12.27550  58.585 

12.27820  63.077 

0-75 
0.8 
0.85 
0.9 
0-95 

10.06040  67.600 
10.06292   72.083 
10.06500  76.564 
10.06645  81.044 

10.06737  85.522 

10.57440  67.591 
10.57680  72.075 
10.57880  76.558 
10.58020  81.040 
10.58110  85.520 

II.II49O   67.582 
II.II72O   72.068 
II.II900   76.553 
II.I2O4O   81.036 
II.I2I2O   85.518 

11.68320  67.574 
11.68540  72.062 
11.68710  76.548 
11.68840  81.032 
11.68920  85.516 

12.28070  67.567 
12.28280  72.056 

12.28440  76.543 

12.28560  81.029 

12.28640  85.515 

1.0 

1-05 
i.i 

1.2 

10.06766  90.000 

10.06737  94.478 

10.06645  98.956 
10.06500  103.436 
10.06292  107.917 

10.58135  90.000 
10.58110  94.480 
10.58020  98.960 
10.57880  103.442 
10.57680  107.925 

II.I2I50   9O.OOO 
II.I2I2O   94.482 
II.I2O4O   98.964 
II.II900  103.447 
II.II72O  .107.932 

11.68946  90.000 
11.68920  94.484 
11.68840  98.968 
11.68710  103.452 
11.68540  107.938 

12.28665  90.000 
12.28640  94.485 
12.28560  98.971 

12.28440  103.457 

12.28280  107.944 

1-25 
1.4 

i-45 

10.06040  112.400 
10.05743  116.886 
10.05410  121.374 
10.05050  125.856 
10.04670  130.360 

10.57440  112.409 
10.57160  116.896 
10.56840  121.386 
10.56500  125.878 
10.56140  130.373 

II.II49O  II2.4I8 
II.II22O  116.906 
II.I092O  121.396 
II.I0600  125.890 
11.10250  130.385 

11.68320  112.426 
11.68060  116.915 
11.67780  121.406 
11.67470  125.900 
11.67140  130.396 

12.28070  112.433 
12.27820  116.923 
12.27550  121.415 
12.27260  125.910 
12.26950  130.406 

i-S 

I.63 
1-65 

10.04280  134.858 
10.03890  139.360 
10.03510  143.865 
10.03150  148.373 
10.02820  152.885 

10.55770  134.871 
10.55400  139.373 
10.55040  143.878 
10.54690  148.385 
10.54380  152.896 

II.O99OO  134.884 
11.09540  139.385 
II.O92OO  143.889 
II.08880  148.396 
11.08580  152.906 

11.66810  134.895 
11.66470  139.396 
11.66140  143.900 
11.65830  148.406 
11.65540  152.915 

12.26630  134.905 
12.26310  139.406 
12.26000  143.909 
12.25700  148.415 
12.25430  152.923 

i!s 
1.85 

1.9 
1.95 

10.02520  157.399  • 
10.02260  161.916 
10.02060  166.435 
10.01910  170.956 
10.01820  175.478 

10.54090  157.409 
10.53850  161.924 

10.53660  166.441 
10.53520  170.960 

10.53430  175-480 

II.083IO  157.417 
II.o8o8o  161.931 
11.07890  166.447 
11.07750  170.964 
11.07670  175.482 

11.65290  157-425 
11.65070  161.938 
11.64890  166.452 
11.64770  170.967 
11.64690  175.484 

12.25180  157.432 

12.24980  161.944 
12.24810  166.457 
12.24690  170.970 

12.24610  175.485 

2.O 

10.01787  180.000 

Jo.53399  180.000 

11.07645   iSo.OOO 

11.64661  180.000 

12.24588  180.000 

Example. 

sinh  (3.2  +  i  i.i)  = 

12.28560  /98°.97i  = 

12.28560  /98°.58'.i6 

t 

[102] 


TABLE  X.    HYPERBOLIC  SINES,     sinh  (x  +  iq)  =  r  /T.    CONTINUED 


=  3.25 


=  3-3 


=  3-35 


*  =  3-4 


*  =  3-45 


o 

12.87578   o.ooo 

13-53788   o.ooo 

14.23382   o.ooo 

14.96536   o.ooo 

I5-73432   o-ooo 

0.05 

12.87600   4.514 

13.53810   4.512 

14.23403  4.511 

14.96556   4.510 

15-73450  4-509 

O.I 

12.87670   9.027 

13.53878  9.024 

14.23470   9.022 

14.96616   9.020 

15.73510   9.018 

0.15 

12.87790  13.539 

13.53991  13.536 

14.23573  13.532 

14.96716  13.529 

15.73603  13-526 

O.2 

12.87949   18.051 

13.54141  18.046 

14.23718  18.041 

14.96855  18.038 

15.73736  18.034 

0.25 

12.88146   22.561 

13.54322  22.555 

14.23897  22.550 

14.97023  22.545 

15.73898  22.541 

o-3 

12.88380  27.070 

13.54550  27.063 

14.24106   27.057 

14.97225  27.052 

15.74088  27.047 

o-3S 

12.88637  31.577 

13.54796  31.569 

14.24339  31.563 

14.97448  31.557 

15.74300  31.551 

0.4 

12.88910  36.082 

13.55062  36.074 

14.24595  36.067 

14.97690  36.061 

15.74529  36.055 

o-45 

12.89215  40.585 

13.55346  40.577 

14.24863  40.570 

14.97945  40.563 

15.74772  40.557 

o-S 

12.89518  45.086 

13-55633  45.078 

14.25137  45.071 

14.98205  45.064 

15.75020  45.058 

0-55 

12.89820  49.585 

13.55918  49-577 

14.25412  49.570 

14.98466  49-563 

15.75270  49-557 

0.6 

12.90117  54.082 

13.56203   54-074 

14.25614  54.067 

14.98721   54.061 

I5-755IO  54-055 

0.65 

12.90398  58.577 

13.56470   58.569 

14.25933  58.563 

14.98960  58.557 

I5.75740  58.551 

0.7 

12.90658  63.070 

13.56718  63.063 

14.26167  63.057 

14.99186  63.052 

I5-75950  63.047 

0-75 

12.90888  67.561 

13.56936   67.555 

14.26377  67.550 

14.99385  67.545 

15.76144  67.541 

0.8 

12.91085   72.051 

I3-57I23   72-046 

14.26556  72.041 

14-99555  72.038 

15-76303   72-034 

0.85 

12.91240  76.539 

I3-57275   76.535 

14.26700  76.532 

14.99692  76.529 

15-76433   76-526 

0.9 

12.91360  81.027 

13-57387  81.024 

14.26805  81.022 

14.99790   81.020 

15-76530  81.018 

o-9S 

12.91430  85.514 

13-57455  85.512 

14.26870  85.511 

14.99853  85.510 

15-76587  85.509 

I.O 

12.91456  90.000 

I3-57476  90.000 

14.26891  90.000 

14.99874  90.000 

15.76607  90.000 

1.05 

12.91430  94.486 

13-57455  94488 

14.26870  94.489 

14.99853  94.490 

15-76587  94-491 

i.i 

12.91360  98.973 

I3-57387  98-976 

14.26805  98.978 

14.99790  98.980 

15-7653°  98-982 

i-iS 

12.91240  103.461 

I3-5727S  103.465 

14.26700  103.468 

14.99692  103.471 

I5-76433  103.474 

1.2 

12.91085  107.949 

13.57123  107.954 

14.26556  107.959 

14.99555  107.963 

15-76303  107.966 

1-25 

12.90888  112.439 

13-56936  112.445 

14.26377  112.450 

14.99385  112.455 

15.76144  112.459 

J-3 

12.90658  116.930 

13.56718  116.937 

14.26167  116.943 

14.99186  116.948 

I5-75950  "6.953 

i-3S 

12.90398  121.423 

13.56470  121.431 

14.25933  121.437 

14.98960  121.443 

I5-75740  121.449 

1.4 

12.90117  125.918 

13.56203  125.926 

14.25614  125.933 

14.98721  125.939 

15.75510  125.945 

i-4S 

12.89820  130.415 

13.55918  130.423 

14.25412  130.430 

14.98466  130.437 

I5-75270  130.443 

J-5 

12.89518  134.914 

I3-55633  I34-922 

14.25137  134.929 

14.98205  134.936 

15.75020  134-942 

i-55 

12.89215  139.415 

13.55346  139-423 

14.24863  139.430 

14.97945  139.437 

15.74772  139-443 

1.6 

12.88910  143.918 

13.55062  143.926 

14.24595  143.933 

14.97690  143.939 

I5.74529  143-945 

i.6S 

12.88637  148.423 

13.54796  148.431 

14.24339  148.437 

14.97448  148.443 

15.74300  148.449 

i-7 

12.88380  152.930 

I3.54550  152.937 

14.24106  152.943 

14.97225  152.948 

15.74088  152.953 

i-75 

12.88146  157.439 

13.54322  157-445 

14.23897  157.450 

14.97023  157.455 

15.73898  157-459 

1.8 

12.87949  161.949 

13.54141  161.954 

14.23718  161.959 

14.96855  161.962 

I5-73736  161.966 

i.8S 

12.87790  166.461 

I3-5399I  166.464 

14.23573  166.468 

14.96716  166.471 

15-73603  166.474 

1.9 

12.87670  170.973 

13.53878  170.976 

14.23470  170.978 

14.96616  170.980 

15.73510  170.982 

i-95 

12.87600  175.486 

1^.53810  175.488 

14.23403  175-489 

14.96556  175.490 

I5-73450  I75-49I 

s.o 

12.87578  180.000 

13.53788  180.000 

14.23382  180.000 

14.96536  180.000 

I5-73432  180.000 

Example.      sinh  (3.3  +  i  1.3)  =  13.56718, 


=  13.56718  /i  i60..s6'.i3*. 


[103] 


TABLE  X.    HYPERBOLIC  SINES,     sinh  (x  +  iq)  =  r  {y.     CONTINUED 


=  3-5 


=  3-55 


x  =  3.6 


*  =  3-65 


=  3-7 


o 

16.54263 

0.000 

17.39230 

o.ooo 

18.28546 

o.ooo 

19.22434 

o.ooo 

20.21129 

o.ooo 

0.05 

16.54281 

4.508 

17.39248 

4-507 

18.28560 

4-507 

19.22448 

4.506 

20.21140 

4.506 

O.I 

16.54337 

9.016 

17.39300 

9.015 

18.28611 

9.013 

19.22496 

9.012 

20.21189 

9.011 

0.15 

16.54428 

13.524 

17.39386 

13-522 

18.28694 

13-519 

19.22577 

13.518 

20.21268 

13-516 

0.2 

16.54550 

18.031 

I7-39504 

18.028 

18.28805 

18.025 

19.22681 

18.023 

20.21365 

18.021 

O.25 

16.54702 

22.537 

17.39650 

22.534 

18.28942 

22.530 

19.22813 

22.527 

20.21490 

22.525 

o-3 

16.54885 

27.042 

17.39822 

27.038 

18.29109 

27-035 

19.22968 

27.031 

20.21639 

27.028 

o-3S 

16.55087 

31-547 

17.40015 

31-542 

18.29292 

31.538 

19.23142 

3I-S3S 

20.21804 

3I-S31 

04 

16.55307 

36.050 

17.40222 

36.045 

18.29490 

36.041 

I9-23331 

36.037 

20.21981 

36-033 

0-45 

16.55538 

40-552 

17.40441 

40.547 

18.29699 

40.542 

I9-23530 

40.538 

20.22170 

40-535 

o-S 

16.55774 

45-052 

17.40665 

45-047 

18.29912 

45-043 

19.23732 

45-039 

20.22365 

45-035 

0-55 

16.56010 

49-552 

17.40900 

49-547 

18.30126 

49.542 

19.23940 

49.538 

20.22560 

49-535 

0.6 

16.56240 

54-050 

17.41110 

54-045 

18.30335 

54.041 

19.24134 

54-037 

20.22750 

54-033 

0.65 

16.56460 

58.547 

17.41310 

58.542 

18.30530 

58.538 

19.24323 

58.535 

20.22930 

58.531 

0.7 

16.56660 

63.042 

17.41510 

63.038 

18.30715 

63-035 

19.24496 

63-031 

20.23090 

63.028 

0-7S 

16.56840 

67-537 

17.41680 

67-533 

18.30880 

67-530 

19.24650 

67-527 

20.23240 

67-525 

0.8 

16.56996 

72.031 

17.41830 

72.028 

18.31020 

72.025 

19.24790 

72.023 

20.23365 

72.021 

0.85 

16.57120 

76.524 

17.41945 

76.521 

18.31130 

76.519 

19.24890 

76.518 

20.23460 

76.516 

0.9 

16.57210 

81.016 

17.42030 

81.015 

18.31210 

81.013 

19.24970 

81.012 

20.23540 

81.011 

o-9S 

16.57260 

85.508 

17.42083 

85-507 

18.31260 

85-507 

19.25015 

85-506 

20.23585 

85.506 

I.O 

16.57282 

90.000 

17.42102 

90.000 

18.31278 

90.000 

19-25033 

90.000 

20.23601 

90.000 

1.05 

16.57260 

94.492 

17.42083 

94-493 

18.31260 

94-493 

19.25015 

94-494 

20.23585 

94.494 

i.i 

16.57210 

98.984 

17.42030 

98.985 

18.31210 

98.987 

19.24970 

98.988 

20.23540 

98.989 

"S 

16.57120 

103.476 

I7-4I945 

103.479 

18.31130 

103.480 

19.24890 

103.482 

20.23460 

103.484 

1.2 

16.56996 

107.969 

17.41830 

107.972 

18.31020 

107.975 

19.24790 

107.977 

20.23365 

107.979 

1-25 

16.56840 

112.463 

17.41680 

112.467 

18.30880 

112.470 

19.24650 

112.473 

20.23240 

"2.475 

i-3 

16.56660 

116.958 

17.41510 

116.962 

18.30715 

116.965 

19.24496 

116.969 

20.23090 

116.972 

1-35 

16.56460 

121.453 

17.41310 

121.458 

18.30530 

121.462 

19.24323 

121.465 

20.22930 

121.469 

1.4 

16.56240 

125.950 

17.41110 

125-955 

18.30335 

125-959 

19.24134 

125.963 

20.22750 

125.967 

i-4S 

16.56010 

130.448 

17.40900 

130.453 

18.30126 

130.458 

19.23940 

130.462 

20.22560 

130.465 

i-5 

16.55774 

134.948 

17.40665 

134-953 

18.29912 

134-957 

19.23732 

134.961 

20.22365 

I34-965 

i-SS 

16.55538 

139.448 

17.40441 

139-453 

18.29699 

I39-458 

19.23530 

139.462 

20.22170 

139-465 

1.6 

16.55307 

I43-950 

17.40222 

143-955 

18.29490 

143-959 

i9-2333i 

143-963 

20.21981 

143.967 

1.65 

16.55087 

148.453 

17.40015 

148.458 

18.29292 

148.462 

19.23142 

148.465 

20.21804 

148.469 

i-7 

16.54885 

152.958 

17.39822 

152.962 

18.29109 

152.965 

19.22968 

152.969 

20.21639 

152.972 

i-7S 

16.54702 

IS7-463 

17.39650 

157.466 

18.28942 

I57-470 

19.22813 

157-473 

20.21490 

157-475 

1.8 

16.54550 

161.969 

17.38504 

161.972 

18.28805 

161.975 

19.22681 

161.977 

20.21365 

161.979 

1.85 

16.54428 

166.476 

17.39386 

166.478 

18.28694 

166.481 

19.22577 

166.482 

20.21268 

166.484 

1.9 

16.54337 

170.984 

17.39300 

170.985 

18.28611 

170.987 

19.22496 

170.988 

20.21189 

170.989 

1-95 

16.54281 

175.492 

17.39248 

175-493 

18.28560 

175-493 

19.2244^ 

175-494 

20.21140 

175-494 

2.O 

16.54263 

180.000 

17.39230 

180.000 

18.28546 

180.000 

19.22434 

180.000 

20.21129 

180.000 

Example.      sinh  (3.65  -+-  to. 25)  =  19.22813  /22°.527  =  19.22813  /22°.3i'.37* 


[104] 


TABLE  X.    HYPERBOLIC   SINES,     sinh  (x  +  iq)  =  r  fy.     CONTINUED 


X  = 

3-75 

x  = 

3-8 

x  = 

3.85 

x  = 

3-9 

*  = 

3-95 

9 

r 

7 

r 

y 

r 

•y 

r 

7 

r 

7 

o 

o 

o 

o 

o 

o 
0.05 

O.I 

0.15 

0.2 

21 
21 

21 
21 
21 

.24878 
.24891 

•24935 
.25006 
.25102 

O.OOO 

4.505 

9.010 

13-514 

18.019 

22.33941 
22.33952 

22-33995 
22.34061 

22.34153 

O.OOO 

4-504 
9.009 

I3-5I3 
I8.0I7 

23.48589 
23.48601 
23.48640 
23.48704 
23.48791 

O.OOO 

4.504 
9.008 

I8.0I5 

24.69110 
24.69120 
24.69159 
24.69223 
24.69302 

O.OOO 

4.504 
9.007 

13-511 

18.014 

25.95806 

25.95820 
25-95854 
25-95911 

25-95991 

O.OOO 

4.503 
9.007 

18.013 

0.25 
0-3 
0-35 
0.4 
0-45 

21 

21 
21 
21 
21 

.25221 
•25362 
.25520 
.25685 
.25869 

22.522 

27.026 

31-528 

36.030 
40.531 

22.34270 
22.34401 
22.34550 
22.34712 
22.34883 

22.52O 
27.023 

3I-526 
36.027 
40.528 

23.48900 
23.49028 

23-49H5 
23.49322 
23.49486 

22.518 
27.O2I 

3I-523 
36-025 
40.526 

24.69406 
24.69528 
24.69662 
24.69809 
24.69964 

22.517 
27.019 
31-521 

36.022 

40.523 

25.96090 
25.96205 

2S-96333 
25.96471 
25.96618 

22.515 
27.017 

31.519 

36.020 

40.521 

o-S 
0-55 
0.6 
0.65 
0.7 

21 

21 
21 
21 
21 

.26052 
.26236 
.26415 
.26586 
.26745 

45-032 
49-531 
54-030 

58.528 

63.026 

22.35060 
22.35230 
22-35403 
22.35565 
22.35716 

45.029 
49.528 
54-027 
58-526 
63.023 

23.49652 
23.49820 
23.49980 
23.50136 
23-50277 

45.026 
49.526 
54-025 

58.523 
63.021 

24.70121 
24.70280 
24.70440 
24.70580 
24.70720 

45.023 
49-523 

54-022 

58.521 

63.019 

25.96770 
25.96920 
25.97066 
25.97206 
25-97337 

45-021 

49-521 

54-020 

58-519 
63.017 

o-75 
0.8 
0.85 
0.9 
o-95 

21 
21 
21 
21 
21 

.26885 
.27000 
.27100 
.27170 
.27212 

67-522 
72.019 
76.514 

81.010 

85.505 

22.35850 
22.35962 
22.36055 
22.36122 
22.36163 

67.520 
72.017 

76-513 
81.009 

85-504 

23.50404 
23-50512 
23.50600 
23.50664 
23.50702 

67.518 
72.015 
76.512 
8I.008 
85.504 

24.70840 
24.70940 
24.71024 
24.71090 
24.71120 

72.014 
76.511 

81.007 

85.504 

25.97450 
25-97550 
25-97630 
25.97680 
25.97720 

67-5I5 
72-013 
76.510 
81.007 
85-503 

I.O 

1.05 
i.i 

1.2 

21 
21 
21 
21 
21 

.27230 
.27212 
.27170 
.27100 
.27000 

90.000 

94-495 
98.990 
103.486 
107.981 

22.36178 
22.36163 
22.36122 
22.36055 
22.35962 

90.000 
94.496 
98.991 
103.487 
107.983 

23-50717 
23.50702 
23.50664 
23.50600 
23-50512 

90.000 
94.496 
98.992 
103.488 
107.985 

24-71135 
24.71120 
24.71090 
24.71024 
24.70940 

90.000 

94.496 
98.993 
103.489 
107.986 

25.9773I 
25.97720 
25.97680 
25.97630 
25-97550 

90.000 

94-497 
98.993 
103.490 
107.987 

1.25 
1.4 

1-45 

21 
21 
21 
21 
21 

.26885 

•26745 
.26586 
.26415 
.26236 

112.478 
116.974 
121.472 
125.970 
130.469 

22.35850 
22.35716 
22.35565 
22.35403 
22.35230 

112.480 
116.977 
121.474 
125-973 
130.472 

23.50404 
23.50277 
23.50136 
23.49980 
23.49820 

112.482 
II6.979 
121.477 

125-975 
130.474 

24.70840 
24.70720 
24.70580 
24.70440 
24.70280 

112.483 
116.981 

121.479 
125.978 
130.477 

25.97450 

25-97337 
25.97206 
25.97066 
25.96920 

112.485 
116.983 
121.481 
125.980 
130.479 

i-SS 
1.6 

-65 
-7 

21 
21 

21 
21 
21 

.26052 
.25869 
.25685 
.25520 
.25362 

134.968 
139.469 
143.970 
148.472 
152.974 

22.35060 
22.34883 
22.34712 
22.34550 
22.34401 

I34.97I 
139.472 

143-973 
148.474 
152.977 

23.49652 
23.49486 
23.49322 

23-49"  5 
23.49028 

134-974 
139-474 
143-975 
148.477 
152.979 

24.70121 
24.69964 
24.69809 
24.69662 
24.69528 

134.976 

139-477 
143.978 
148.479 

152.981 

25.96770 
25.96618 
25.96471 

25.96205 

134-979 
139-479 
143.980 
148.481 
152.983 

•75 
.8 

•85 
•9 
1-95 

21 
21 
21 
21 
21 

.25221 
.25102 
.25006 

•24935 
.24891 

I57-478 
161.981 
166.486 
170.990 
175-495 

22.34270 
22.34153 
22.34061 

22-33995 
22.33952 

157.480 
161.983 
166.487 
170.991 
175.496 

23.48900 
23.48791 
23.48704 
23.48640 
23.48601 

157.482 
161.985 
166.488 
I7O.992 
175.496 

24.69406 
24.69302 
24.69223 
24.69159 
24.69120 

157.483 
161.986 
166.489 
170.993 
175.496 

25.96090 
25-9599I 
25-959" 
25-95854 
25.95820 

I57.485 
161.987 
166.490 
170.993 
175-497 

2.O 

21.24878 

180.000 

22.33941 

180.000 

23.48589 

iSo.OOO 

24.69110 

180.000 

25.95806 

180.000 

Example. 

sinh  (3.90  +  *  1.90)  = 

24.69159 

/I70°.993  =  24-69159  /I70°.59' 

35'. 

[105] 


TABLE  XI.     HYPERBOLIC   COSINES,     cosh  (*  +  ig)  =  r  /y 


x  =  0.05 


x,  =  0.15 


o 

0.05 

O.I 

0.15 

O.2 

I.OOOOO  O.OO 

0.99692  o.oo 
0.98769  o.oo 
0.97237  o.oo 
0.95106  o.oo 

1.00125   o-00 
0.99817   0.225 
0.98895   0.453 
0.97366   0.687 
0.95237   0.930 

1.00500   o.oo 
1.00194   0.450 
0.99275   0.905 
0.97752   1.371 
0.95632   1.855 

1.01127   o.oo 
1.00822    0.671 

0.99909   1.351 
0.98396   2.047 
0.96290   2.769 

1.02007   o.oo 
1.01704   0.890 
1.00800   1.791 
0.99300   2.713 
0-97213   3-669 

O.25 
0-3 
0-35 
0.4 

0-45 

0.92388  o.oo 
0.89101  o.oo 
0.85264  o.oo 
0.80902  o.oo 
0.76041  o.oo 

0.92523   1.186 
0.89237   1.458 
0.85407   1.754 
0.81053   2.079 
0.76202   2.443 

0.92930   2.364 
0.89662   2.907 
0.85851   3.495 
0.81520   4.142 
0.76698   4.866 

0.93607   3.529 
0.90364  4.338 
0.86583   5.213 

0.82291   6.174 

0.77527   7.247 

0.94556   4-674 
0.91347   5-743 
0.87609   6.897 
0.83370   8.161 
0.78661   9-569 

o-5 

0.55 

0.6 
0.65 
0.7 

0.70711  o.oo 
0.64945  o.oo 
0.58779  o.oo 
0.52250  o.oo 
0-45399  0-0° 

0.70803   2.860 

0-65135   3-348 
0.58989   3.934 
0.52487   4.661 
0.45672   5.600 

0.71417   5.692 
0.65713   6.656 
0.59626   7.811 
0.53201   9.238 
0.46491   1  1.  068 

0.72296   8.468 

0.66667   9-889 

0.60676   11.581 

0.54376  13.656 

0.47831   16.289 

0.73521   11.165 
0.67994  13.012 
0.62131   15.198 

0-55995   17-853 
0.49663   21.175 

o-7S 
0.8 
0.85 
0.9 
0-95 

0.38268  o.oo 
0.30902  o.oo 
0.23345  o.oo 
0.15643  o.oo 
0.07846  o.oo 

0.38594   6.877 
0.31304   8.741 
0.23874   11.755 
0.16424  17.507 
0.09305  32.407 

0.39558   13-529 
0.32485   17-053 
0.25403   22.545 
0.18576  32.181 
0.12724  51.700 

0.41124  19.771 

0.34375  24.618 
0.27779  3^805 

0.21712  43.229 
0.16978  62.139 

0.43242   25.478 
0.36882  31.277 
0.30827  39.424 

0.25497  5I-254 
0.21608  68.261 

I.O 

1.05 
i.i 

1.2 

o.oo    180 
0.07846  180 
0.15643  180 
0.23345  180 
0.30902  1  80 

0.05002  90.000 
0.09305  147-593 
0.16424  162.493 
0.23874  168.245 
0.31304  171.259 

0.10017  90.000 
0.12724  128.300 
0.18576  147.819 
0.25403  157-455 
0.32485  162.947 

0.15056   90.000 
0.16978  117.861 

0.21712  136.771 
0.27779  148.195 
0.34375  155.382 

0.20134  90.000 
0.21608  111.739 
0.25497  128.746 
0.30827  140.576 
0.36882  148.723 

1-25 

i-35 
1.4 

MS 

0.38268  180 
0-45399  180 
0.52250  180 
0.58779  180 
0.64945  1  80 

0.38594  173-123 
0.45672  174.400 

0.52487  175-339 
0.58989  176.066 
0.65135  176.652 

0.39558  166.471 
0.46491  168.932 
0.53201  170.762 
0.59626  172.189 
0.65713  173-344 

0.41124  160.229 
0.47831  163.711 
0.54376  166.344 

0.60676  168.419 
0.66667  170.111 

0.43242  154.522 
0.49663  158.825 
0-55995  162.147 
0.62131  164.802 
0.67994  166.988 

i-SS 
1.6 
1.65 

0.70711  180 
0.76041  1  80 
0.80902  1  80 
0.85264  1  80 
0.89101  180 

0.70803  177-139 
0.76202  177.557 
0.81053  177.921 
0.85407  178.246 
0.89237  178.542 

0.71417  174.308 
0.76698  175.134 
0.81520  175.858 
0.85851  176.505 
0.89662  177.093 

0.72296  171.532 
0.77527  172.753 

0.82291  173.826 
0.86583  174.787 
0.90364  175.662 

0.73521  168.835 
0.78661  170.431 
0.83370  171-839 
0.87609  173.103 
0.91347  174-257 

i-75 
1.8 

1.85 
1.9 
i-95 

0.92388  180 
0.95106  180 
0.97237  180 
0.98769  180 
0.99692  180 

0.92523  178.814 
0.95237  179.070 
0.97366  179-313 
0.98895  179.547 
0.99817  179-775 

0.92930  177.636 
0.95632  178.145 
0.97752  178.629 
0.99275  179.095 
1.00194  179.550 

0.93607  176.471 
0.96290  177.231 

0.98396  177.953 

0.99909  178.649 
1.00822  179.329 

0.94556  175-326 
0.97213  176.331 
0.99300  177.287 
1.00800  178.209 
1.01704  179.110 

2.O 

i.ooooo  180 

1.00125  180.000 

1.00500  180.000 

1.01127  180.000 

1.02007  180.000 

Example. 

cosh  (o.io  +  10.55)  =  0.65713  /6°.6^6  =  0.65713  /6°.39 

'.22*. 

[106] 


TABLE  XI.     HYPERBOLIC   COSINES,     cosh  (x  +  ig)  =  r  Ay.     CONTINUED 


X  = 

0.25 

x  = 

o-3 

x  = 

0-35 

x  = 

0.4 

x  — 

0.45 

9 

r 

7 

r 

y 

r 

y 

r 

y 

r 

y 

0 

o 

o 

o 

o 

0 

1.03141 

o.ooo 

1-04534 

o.ooo 

i.  06  1  88 

o.ooo 

1.08107 

O.OOO 

1.10297 

O.OOO 

0.05 

1.02843 

1.104 

1.04239 

1-313 

1.05898 

1.516 

1.07822 

I.7I3 

1.10017 

I.9O2 

O.I 

1.01949 

2.221 

1-03357 

2.642 

1.05029 

3-050 

1.06970 

3-444 

1.09182 

3-823 

0.15 

1.00464 

3-365 

1.01894 

4.001 

1.03590 

4.617 

1-05557 

5-212 

1.07798 

5-784 

O.2 

0.98403 

4-550 

0.99862 

5-407 

1.01592 

6.238 

1-03597 

7.038 

1.05880 

7.806 

0.25 

0-95779 

5-793 

0.97277 

6.881 

0.99052 

7-932 

1.01107 

8.944 

1.03446 

9-9I3 

°-3 

0.92612 

7-ii3 

0.94161 

8.443 

0.95994 

9.726 

0.98113 

10.957 

1.00521 

12.132 

o-35 

0.88927 

8-536 

0-90539 

IO.I22 

0.92444 

11.647 

0.94642 

13.107 

0.97136 

14.496 

0.4 

0.84754 

10.090 

0.86443 

11.951 

0.88436 

13-733 

0.90732 

I5-432 

0-93330 

17.042 

0-45 

0.80127 

11.815 

0.81912 

I3-972 

0.84012 

16.029 

0.86426 

17.979 

0.89149 

I9.8l6 

0-5 

0.75088 

13.762 

0.76989 

16.241 

0.79220 

18.592 

0.81775 

20.804 

0.84469 

22.875 

o-55 

0.69685 

16.001 

0.71730 

18.834 

0.74119 

21-497 

0.76844 

23.982 

0.79895 

26.288 

0.6 

0.63977 

18.629 

0.66199 

2  1  .849 

0.68781 

24.843 

0.71708 

27.607 

0.74969 

30.144 

0.65 

0.58036 

21.785 

0.60476 

25-425 

0.63292 

28.763 

0.66462 

31.800 

0.69968 

34-546 

0.7 

0.51954 

25-673 

0.54666 

29-758 

0.57766 

33-!57 

0.61223 

36.712 

0.65012 

39.626 

0-75 

0-45854 

30.595 

0.48906 

35-II8 

0.52348 

39-079 

0.56140 

42.529 

0.60249 

45-527 

0.8 

o-399J3 

37.008 

0.43385 

41-878 

0.47231 

45-992 

0.51401 

49.464 

0.55860 

52-399 

0.85 

0.34396 

45-057 

0.38370 

50.507 

0.42671 

54-484 

0.47246 

57-712 

0.52062 

60.358 

0.9 

0.29782 

57.110 

0-34235 

61.467 

0.38994 

64.786 

0-43953 

67-37I 

0.49093 

69.424 

0-95 

0.26452 

72.186 

0.31447 

74.882 

0.36571 

76.831 

0.41818 

78.297 

0.47191 

79-433 

I.O 

0.25261 

90.000 

0.30452 

90.000 

0.35719 

90.000 

0.41075 

90.000 

0.46534 

90.000 

1.05 

0.26452 

107.814 

0.31447 

I05.II8 

0.36571 

103.169 

0.41818 

101.703 

0.47191 

100.567 

i.i 

0.29782 

122.890 

0.34235 

"8.533 

0.38994 

115.214 

0-43953 

112.629 

0.49093 

110.576 

US 

0.34396 

134-943 

0.38370 

129.493 

0.42671 

125.516 

0.47246 

122.288 

0.52062 

119.642 

1.2 

0.39913 

142.992 

0-43385 

138.122 

0.47231 

134.008 

0.51401 

130.536 

0.55860 

127.601 

1-25 

0.45854 

149.405 

0.48906 

144.882 

0.52348 

140.921 

0.56140 

I37.47I 

0.60249 

134-473 

i-3 

0.51954 

I54-327 

0.54666 

150.242 

0.57766 

146.843 

0.61223 

143-288 

0.65012 

140.374 

i-3S 

0.58036 

158.215 

0.60476 

154-575 

0.63292 

151-237 

0.66462 

148.200 

0.69968 

145-454 

1.4 

0.63977 

161.371 

0.66199 

I58.I5I 

0.68781 

I55.I57 

0.71708 

152.393 

0.74969 

149-856 

i-45 

0.69685 

163.999 

0.71730 

l6l.l66 

0.74119 

158.503 

0.76844 

156.018 

0.79895 

I53.7I2 

i-5 

0.75088 

166.238 

0.76989 

163.759 

0.79220 

161.408 

0.81775 

159.196 

0.84649 

157-125 

i-5S 

0.80127 

168.185 

0.81912 

166.028 

0.84012 

163.971 

0.86426 

162.021 

0.89149 

160.184 

1.6 

0.84754 

169.910 

0.86443 

168.049 

0.88436 

166.267 

0.90732 

164.568 

0.93330 

162.958 

1.65 

0.88927 

171.464 

0-90539 

169.878 

0.92444 

168.353 

0.94642 

166.893 

0.97136 

165.504 

i-7 

0.92612 

172.887 

0.94161 

I7L557 

0.95994 

170.274 

0.98113 

169.043 

1.00521 

167.868 

1-75 

0-95779 

174.207 

0.97277 

I73-H9 

0.99052 

172.068 

1.01107 

171.056 

1.03446 

170.087 

1.8 

0.98403 

I75-450 

0.99862 

174-593 

1.01592 

173.762 

1-03597 

172.962 

1.05880 

172.194 

1.85 

i  .00464 

176.635 

1.01894 

175-999 

1.03590 

175-383 

1-05557 

174-788 

1.07798 

174.216 

1.9 

1.01949 

177.779 

1-03357 

177.358 

1.05029 

176.950 

1.06970 

176.556 

1.09182 

176.177 

i-9S 

1.02843 

178.896 

1.04239 

178.687 

1.05898 

178.484 

1.07822 

178.287 

1.10017 

178.098 

2.0 

1.03141 

180.000 

1-04534 

180.000 

1.06188 

180.000 

1.08107 

180.000 

1.10297 

180.000 

Example.       cosh  (0.40  +  i  0.5)  =  0.81775 


=  0.81775 


[107] 


TABLE  XI.    HYPERBOLIC  COSINES,     cosh  (x  +  iq)  =  r  /y.    CONTINUED 


*  = 

o.S 

X  = 

0-55 

x  = 

0.6 

x  = 

0.65 

x  = 

0.7 

q 

r 

7 

r 

7 

r 

7 

r 

7 

r 

7 

o 

0 

o 

o 

0 

o 

1.12763 

o.ooo 

i.iSSio 

o.ooo 

1.18547 

o.ooo 

1.21879 

O.OOO 

1-25517 

o.ooo 

0.05 

1.12489 

2.083 

1.15244 

2.256 

1.18287 

2.420 

1.21626 

2.576 

1.25271 

2.723 

O.I 

1.11672 

4.186 

1.14446 

4-533 

1.17510 

4.879 

1.20871 

5-J74 

1-24538 

5.468 

0.15 

1.10320 

6.331 

1.13127 

6.852 

1.16226 

7-347 

1.19623 

7-815 

1.23327 

8.256 

O.2 

1.08446 

8.539 

1.11300 

9-237 

1.14448 

9.898 

1.17892 

10.523 

1.21653 

II.  IIO 

0.25 

1.06070 

10.836 

1.08987 

11.713 

I.I22OO 

12.541 

I-I57I5 

13.322 

1.19541 

14.054 

o-3 

1.03220 

13.249 

1.06214 

14-307 

1.09509 

15-304 

1.13108 

16.240 

1.17019 

17.116 

0-35 

0.99927 

15.811 

1.03004 

17.052 

I.064II 

18.216 

I.IOIII 

19.306 

1.14125 

20.323 

0.4 

0.96232 

18.559 

0-99437 

19.984 

1.02948 

21-315 

1.06769 

22.555 

1.10904 

23.706 

0-45 

0.92182 

21.538 

0.95524 

23.146 

0.99174 

24.640 

1-03134 

26.024 

1.07409 

27.302 

o-S 

0.87837 

24.803 

0.91338 

26.589 

0.95149 

28.238 

0.99270 

29-755 

1.03704 

31.148 

o-SS 

0.83266 

28.416 

0.86951 

30.372 

0.90946 

32.162 

0.95249 

33-796 

0.99861 

35.284 

0.6 

0.78552 

32-458 

0.82447 

34-563 

0.86650 

36.471 

0.91156 

38.197 

0.95966 

39-755 

0.65 

0-73793 

37.020 

0.77927 

39-241 

0.82361 

41.231 

0.87090 

43-on 

0.92112 

44.603 

0.7 

0.69112 

42.207 

0.73510 

44.489 

0.78194 

46.506 

0.83160 

48.290 

0.88406 

49.867 

0-75 

0.64652 

48.129 

0.69333 

50.390 

O.74282 

52.358 

0.79492 

54-074 

0.84965 

55-574 

0.8 

0.60583 

54.889 

0.65555 

57.010 

0.70769 

58.826 

0.76220 

60.387 

0.81911 

61.737 

0.85 

0.57100 

62.547 

0.62350 

64-375 

0.67810 

65.914 

0.73482 

67.219 

0.79369 

68-335 

0.9 

0.54407 

71.082 

0.59894 

72.441 

0.65559 

73-568 

0.71409 

74.5I4 

0-77455 

75.3I5 

o.QS 

0.52697 

80.335 

0.58345 

81.064 

0.64147 

81.663 

0.70115 

82.161 

0.76263 

82.581 

I.O 

0.52110 

90.000 

0.57815 

90.000 

0.63665 

90.000 

0.69675 

90.000 

0.75858 

90.000 

1.05 

0.52697 

99.665 

0.58345 

98-936 

0.64147 

98.337 

0.70115 

97.839 

0.76263 

97.419 

i.i 

0.54407 

108.918 

0.59894 

107-559 

0.65559 

106.432 

0.71409 

105.486 

0-77455 

104.685 

i-iS 

0.57100 

117.453 

0.62350 

115.625 

0.67810 

114.086 

0.73482 

112.781 

0.79369 

111.665 

1.2 

0.60583 

125.111 

0.65555 

122.990 

0.70769 

121.174 

0.76220 

119.613 

0.81911 

118.263 

I.2S 

0.64652 

131.871 

0.69333 

129.610 

0.74282 

127.642 

0.79492 

125.926 

0.84965 

124.426 

i-3 

0.69112 

137.793 

o.735io 

I35-51! 

0.78194 

133-494 

0.83160 

131.710 

0.88406 

130.133 

i-35 

0-73793 

142.980 

0.77927 

140.759 

0.82361 

138.769 

0.87090 

136.989 

0.92112 

135-397 

1.4 

0.78552 

147.542 

0.82447 

145-437 

0.86650 

I43-529 

0.91156 

141.803 

0.95966 

140.245 

i.  45 

0.83266 

151-584 

0.86951 

149.628 

0.90946 

147.838 

0.95249 

146.204 

0.99861 

144.716 

1.50 

0.87837 

155.198 

0.91338 

153-4" 

0.95149 

151.762 

0.99270 

150.245 

1.03704 

148.853 

i-SS 

0.92182 

158.462 

0.95524 

156.854 

0.99174 

155-360 

1-03134 

I53.976 

1.07409 

152.698 

i.  60 

0.96232 

161.441 

0-99437 

160.016 

I.O2948 

158.685 

1.06769 

157-445 

1.10904 

156-294 

1.65 

0.99927 

164.189 

1.03004 

162.948 

1.06411 

161.784 

I.IOIII 

160.694 

1.14125 

159.677 

1.70 

1.03220 

166.751 

1.06214 

165.693 

1.09509 

164.696 

1.13108 

163.760 

1.17019 

162.884 

i-75 

1.06070 

169.164 

1.08987 

168.287 

.I22OO 

167.459 

1.15715 

166.678 

1.19541 

165.946 

1.8 

i  .08446 

171.461 

1.11300 

170.763 

.14448 

170.102 

1.17892 

169.477 

1.21653 

168.890 

1.85 

1.10320 

173.669 

1.13127 

173.148 

.16226 

172.653 

1.19623 

172.185 

1.23327 

171.744 

i.  9 

1.11672 

175.814 

1.14446 

175467 

.17510 

175.121 

1.20871 

174.826 

1-24538 

I74-532 

i-9S 

1.12489 

177.917 

1.15244 

177-744 

.18287 

177.580 

1.21626 

177.424 

1.25271 

177.277 

2.0 

1.12763 

180.000 

1.15510 

180.000 

I.I8547 

180.000 

1.21879 

180.000 

1-25517 

180.000 

Example.       cosh  (0.65  +  i  i.o)  =  0.69675  /go0. 


[108] 


TABLE  XI.     HYPERBOLIC   COSINES,     cosh  (x  +  iq)  =  r  fy.     CONTINUED 


x  =  0.75 


=  0.85 


x  =  o.g 


x  =  0.95 


o 

1.29468 

0.000 

1  -33  743 

o.ooo 

I-38353 

o.ooo 

1.43309 

o.ooo 

1.48623 

o.ooo 

0.05 

1.29230 

2.862 

I-335I3 

2.992 

1.38130 

3-"3 

1.43094 

3.227 

1.48415 

3-332 

O.I 

1.28520 

5-745 

1.32825 

6.004 

1.37466 

6.246 

1.42452 

6.473 

1.47797 

6.684 

0.15 

1.27346 

8.670 

1.31691 

9-058 

1.36369 

9.420 

I-4I395 

9-758 

1.46778 

10.071 

O.2 

1.25726 

11.661 

1.30124 

12.175 

1-34858 

12.656 

1-39937 

13.102 

1.45371 

13-516 

0.2S 

1.23689 

14.740 

1.28152 

15-379 

I-32955 

15-974 

1.38105 

16.526 

1.43611 

17.037 

o-3 

1.21248 

17-933 

1.25802 

18.693 

1.30693 

19.398 

I-35927 

20.051 

1.41519 

20.653' 

0.35 

1.18457 

21.267 

1-23115 

22.143 

1.28107 

22.952 

1-33444 

23.699 

1.39136 

24-387 

0.4 

1-15356 

24.772 

1.20135 

25-755 

1.25246 

26.601 

1.30700 

27.493 

1-36505 

28.257 

0-45 

I.I200I 

28.478 

1.16917 

29-559 

1.22163 

30-550 

1.27748 

31.457 

1.33682 

32.286 

o-S 

1.08453 

32-442 

1.13522 

33-586 

1.18918 

34.647 

1.24649 

35-614 

1.30723 

36-493 

0-55 

1.04785 

36-637 

1.10024 

37.865 

1-15583 

38.978 

1.21471 

39.986 

1.27697 

40.898 

0.6 

I.OI079 

41.160 

1.06500 

42.426 

1.12234 

43-567 

1.18289 

44-593 

1.24674 

45-517 

0.65 

0.97427 

46.026 

1.03041 

47.298 

1.08957 

48.435 

1.15184 

49-453 

1.21732 

50.363 

0.7 

0.93932 

51-263 

0.99742 

52-500 

1.05843 

53-599 

1.12243 

54-574 

1.19227 

55-443 

0-7S 

O.9O7OO 

56.889 

0.96705 

58.045 

1.02985 

59.062 

1-09553 

59.961 

1.16418 

6o.755 

0.8 

0.87846 

62.907 

0.94033 

63.927 

1.00481 

64.819 

1.07202 

65.601 

1.14209 

66.288 

0.85 

0.85481 

69.294 

0.91828 

70.123 

0.98420 

70.842 

1.05273 

71.471 

1.12400 

72.020 

0.9 

0.83  706 

75-998 

0.90178 

76.585 

0.96883 

77.091 

1-03837 

77-532 

1.11056 

77-9I5 

0-95 

0.82605 

82.936 

0.89157 

83.241 

0-95933 

83-503 

1.02951 

83-730 

1.10227 

83.927 

I.O 

O.82232 

90.000 

0.88811 

90.000 

0.95612 

90.000 

1.02652 

90  ooo 

1.09948 

90.000 

1.05 

0.82605 

97.064 

0.89157 

96.759 

0-95933 

96.497 

1.02951 

96.270 

1.10227 

96.073 

i.i 

0.83706 

104.002 

0.90178 

103.415 

0.96883 

102.909 

1-03837 

102.468 

1.11056 

102.085 

I.IS 

0.85481 

110.706 

0.91828 

109.877 

0.98420 

109.158 

1.05273 

108.529 

I  12400 

107.980 

1.2 

0.87846 

117.093 

0.94033 

116.073 

1.00481 

115.181 

1.07202 

114.399 

1.14209 

113.712 

1.  25 

0.90700 

123.111 

0.96705 

121.955 

1.02985 

120.938 

1-09553 

1  20.039 

1.16418 

119.245 

i-3 

0.93932 

128.737 

0.99742 

127.500 

1.05843 

126.401 

1.12243 

125.426 

1.19227 

124-557 

i-35 

0.97427 

133-974 

1.03041 

132.702 

1.08957 

i3!-565 

1.15184 

130.547 

1.21732 

129.637 

1.4 

I.OI079 

138.840 

1.06500 

137-574 

1.12234 

136.433 

1.18289 

135.407 

1.24674 

134-483 

1-45 

1.04785 

I43-363 

1.10024 

142-135 

1-15583 

141.022 

1.21471 

140.014 

1.27697 

139.102 

i-5 

1.08453 

I47-578 

1.13522 

146.414 

1.18918 

145-353 

1.24649 

144.386 

1.30723 

I43-507 

i-SS 

I.I2OOI 

151-522 

1.16917 

150.441 

1.22163 

149.450 

1.27748 

148.543 

1.33682 

147.714 

1.6 

I-I5356 

155.228 

1.20135 

I54-24S 

1.25246 

153-399 

1.30700 

152.507 

1-36505 

I5I-743 

1.65 

1.18457 

158.733 

i.23"5 

I57-857 

1.28107 

157.048 

1.33444 

156.301 

1.39136 

I55-6I3 

i-7 

I.2I248 

162.067 

1.25802 

161.307 

1.30693 

160.602 

I-35927 

159-949 

1.41519 

159-347 

i-75 

1.23689 

165.260 

1.28152 

164.621 

I-32955. 

164.026 

1.38105 

163.474 

1.43611 

162.963 

1.8 

1.25726 

168.339 

1.30124 

167.825 

1-34858 

167.344 

1-39937 

166.898 

1.45371 

166.484 

1.85 

1.27346 

171-330 

1.31691 

170.942 

1.36369 

170.580 

I-4I395 

170.242 

1.46778 

169.929 

1.9 

I.2852O 

174-255 

1.32825 

173.996 

1.37466 

173-754 

1.42452 

173-527 

1.47797 

I73-3I6 

i  -95 

1.29230 

177.138 

I-335I3 

177.008 

1.38130 

176.887 

1.43094 

176.773 

1.48415 

176.668 

2.O 

1.29468 

180.000 

1-33743 

180.000 

I-38353 

180.000 

1-43309 

180.000 

148623 

180.000 

Example. 

cosh  (0.90  +  *  0.5) 

=  1.24649  /35°-6i4 

=  1.24649  /35°-36'.5o'. 

[109] 


TABLE  XI.    HYPERBOLIC   COSINES,     cosh  (at  +  iq)  =  r  /y.     CONTINUED 


x  =  1.05 


x  =  1.15 


o 
0.05 

O.I 

0.15 

O.2 

1.54308  o.ooo 

1.54108  3.276 
1.53513  6.878 
1.52532  10.362 
1.51182  13.899 

1.60379  o.ooo 
1.60187  3.521 
1.59614  7.059 
1.58671  10.631 
1-57374  14-253 

1.66852  o.ooo 
1.66667  3-605 
1.66117  7.226 
1.65211  10.879 

1.63965  14.579 

1.73741  o.ooo 

1.73564  3-683 
1-73036  7-380 

1.72166  11.107 
1.70971  14.880 

i.  81066  o.ooo 

1.80895  3.754 

1.80389  7.522 
1-79555  11-318 
1.78409  15.156 

0.25 

0-3 
0-35 
0.4 

0-45 

1.49487  17.509 
1.47479  21.209 
1.45193  25.019 
1.42675  28.957 
1.39976  33-043 

1-55747  17-944 
1.53820  21.720 
1.51630  25.599 
1.49220  29.597 
1.46641  33-732 

1.62404  18.344 
1.60558  22.189 
1.58460  26.130 
1.56156  30.182 

1.53694  34-360 

1.69472  18.713 

1.67705  22.620 
1.65699  26.616 

1-63497  30.716 
1.61147  34.931 

1.76975  19.050 
1.75282  23.014 
1-73363  27.061 
1.71260  31.203 
1.69018  35-451 

o-5 
0-55 
0.6 
0.65 

0.7 

1.37153  37.293 
1.34272  41.724 
1.31400  46.349 

1.28612  51.179 

1.25984  56.216 

1.43950  38.019 
1.41207  42.470 
1.38479  47-098 
I-35837  5i-910 
I-3335I  56-907 

1.51128  38.677 

1.48517  43-H5 
1.45926  47-773 
1.43421  52-565' 
1.41070  57-523 

1.58701  39.275 

1.56217  43-755 
1.53756  48.380 
1.51381  53-J53 
i-49I55  58-074 

1.66688  39.816 
1.64325  44-307 
1.61987  48.927 
1-59733  53-681 
1.57626  58.567 

0-75 
0.8 
0.85 
0.9 
o-95 

1.23594  61.459 
1.21515  66.895 

1.19816  72.504 

1.18552  78.252 
1.17782  84.100 

1.31095  62.085 
1.29137  67.432 
1.27540  72.929 
1.26358  78.547 
1.25631  84.252 

1.38939  62.641 
1.37093  67.908 
1-35590  73-306 
1.34478  78.808 
1-33795  84.385 

1.47141  63.'i36 
1.45399  68.330 
1-43983  73.639 
1.42936  79-039 
1.42294  84.503 

1.55721  63.579 
1.54077  68.706 
1.52741  73.935 

I.5I755  79-243 
1.51150  84.607 

I.O 

1.05 
i.i 
i-iS 

1.2 

1.17520  90.000 

1.17782  95.900 
1.18552  101.748 

1.19816  107.496 
1.21515  113.105 

1.25386  90.000 
1.25631  95.748 
1.26358  101.453 
1.27540  107.071 
1.29137  112.568 

1  -33565  90-0°o 
1-33795  95-6i5 
1.34478  101.192 
1.35590  106.694 
1.37093  112.092 

1.42078  90.000 
1.42294  95.497 
1.42936  100.961 
1.43983  106.361 
1.45399  111-670 

1.50946  90.000 
1.51150  95.393 
1.51755  100.757 
1.52741  106.065 
1.54077  111.294 

1-25 

i-3 
i-35 
1.4 

i-45 

1.23594  118.541 
1.25984  123.784 
1.28612  128.821 
1.31400  133.651 
1.34272  138.276 

1.31095  117.915 
1.33351  123.093 
1.35837  128.090 
1.38479  132.902 
1.41207  137.530 

1.38939  H7-359 
1.41070  122.477 
1.43421  127.435 
1.45926  132.227 
1.48517  136.855 

1.47141  116.864 
1.49155  121.926 
1.51381  126.847 
J-53756  131.620 
1.56217  136.245 

1.55721  116.421 
1.57626  121.433 
1.59733  126.319 
1.61987  131.073 
1.64325  135.693 

i-5 
i-SS 
1.6 

1.65 
i-7 

1.37153  142.707 
1.39976  146.957 
1.42675  151.043 
1.45193  154.981 

1.47479  I58.791' 

1.43950  141.981 
1.46641  146.268 
1.49220  150.403 
1.51630  154-401 
1.53820  158.280 

1.51128  141-323 
1.53694  145.640 
1.56156  149.818 
1.58460  1.53.870 
1.60558  157.811 

1.58701  140.725 
1.61147  145.069 
1.63497  149.284 
1.65699  153-384 
1.67705  157-380 

1.66688  140.184 
1.69018  ^44-549 
1.71260  148.797 

I.73363  152.939 
1.75282  156.986 

i-7S 
1.8 
1.85 
1.9 
I-9S 

1.49487  162.491 
1.51182  166.101 
1.52532  169.638 
i-SSS^  173-122 
1.54108  176.724 

1.55747  162.056 
1.57374  165.747 
1.58671  169.369 
1.59614  172.941 
1.60187  176.479 

1.62404  161.656 
1.63965  165.421 
1.65211  169.121 
1.66117  172.774 
1.66667  176.395 

1.69472  161.287 
1.70971  165.120 
1.72166  168.893 
1.73036  172.620 
1.73564  176.317 

1.76975  160.950 
1.78409  164.844 
1-79555  168.682 
1.80389  172.478 
1.80895  176.246 

2.0 

1.54308  180.000 

1.60379  180.000 

1.66852  180.000 

1.73741  180.000 

i.  81066  180.000 

Example.       cosh  (1.20  +  io)  =  i. 81066  /o°. 


[no] 


TABLE  XI.    HYPERBOLIC  COSINES,     cosh  (x  +  iq)  =  r  /T.    CONTINUED 


x  =  1.25 


x  =  1.3 


=  i-3S 


=  1.4 


=  1-45 


o 

1.88842 

o.ooo 

1.97091 

o.ooo 

2.05833 

o.ooo 

2.15090 

O.OOO 

2.24884 

0.000 

0.05 

1.88679 

3.820 

1.96935 

3.880 

2.05684 

3-935 

2.14947 

3.986 

2.24747 

4.032 

O.I 

I.88I93 

7.652 

1.96470 

7.772 

2.05238 

7.882 

2.14520 

7.982 

2-24334 

8.075 

0.15 

1.87394 

11.511 

1.95704 

11.689 

2.04505 

11.851 

2.13819 

I2.OOO 

2.23669 

12.136 

O.2 

1.86297 

15.410 

1.94654 

15.642 

2.03500 

15-855 

2.12858 

16.049 

2.22751 

16.226 

0.25 

1.84924 

19.360 

1.93341 

19.644 

2.02245 

19.903 

2.11658 

20.139 

2.21604 

20.355 

°-3 

1.83304 

23-375 

1.91792 

23.705 

2.00764 

24.006 

2.10244 

24.281 

2.20254 

24.531 

o-35 

I.8I4/O 

27.467 

1  .90040 

27.837 

1.99091 

28.175 

2.08647 

28.482 

2.18731 

28.762 

0.4 

1.79462 

31.646 

1.88123 

32.050 

1.972612 

32-417 

2.06903 

32.751 

2.17067 

33.055 

0-45 

I-77323 

35-923 

1.86084 

26.352 

1-95319 

36.742 

2.05051 

37-095 

2.15302 

37.416 

o-S 

I.75I04 

40.308 

1.83970 

40.752 

1.93306 

4i-i55 

2.03135 

41.520 

2.13478 

41.851 

0-55 

1.72856 

44.805 

1.81832 

45.255 

1.91273 

45.662 

2.OI2OO 

46.030 

2.11637 

46.362 

0.6 

I-70635 

49.420 

1.79722 

49.865 

1.89268 

50.265 

1.99295 

50.627 

2.09828 

50.953 

0.65 

1.68497 

54-155 

1.77694 

54.582 

1.87343 

54.96S 

1.97468 

55-3" 

2.08093 

55.621 

0.7 

I.6650I 

59.009 

1.75802 

59.405 

1.85549 

59.760 

1.95767 

60.079 

2.06479 

60.366 

0-75 

1.64699 

63.974 

1.74096 

64.327 

1.83934 

64.644 

I.94237 

64.927 

2.05030 

65.182 

0.8 

I-63I45 

69.042 

1.72627 

69.341 

1.82544 

69.608 

I.9292I 

69.847 

2.03784 

70.061 

0.85 

1.61884 

74-197 

1-71435 

74-432 

1.81417 

74.641 

1.91856 

74.828 

2.02775 

74-995 

0.9 

1.60954 

79.424 

1-70557 

79-585 

1.80588 

79.729 

I.9I072 

79.857 

2.02034 

79.972 

0-95 

1.60384 

84.699 

1.70019 

84.782 

1  .80080 

84-855 

1.90592 

84.920 

2.01580 

84.979 

I.O 

1.60192 

90.000 

1.69838 

90.000 

1.79909 

90.000 

1.90430 

90.000 

2.01427 

90.000 

1.05 

1.60384 

95-301 

1.70019 

95.218 

1  .80080 

95-145 

1.90592 

95.080 

2.01580 

95.021 

i.i 

1.60954 

100.576 

1-70557 

100.415 

1.80588 

100.271 

I.9IO72 

100.143 

2.02034 

100.028 

i-iS 

1.61884 

105.803 

1-71435 

105.568 

1.81417 

105-359 

1.91856 

105.172 

2.02775 

105.005 

1.2 

I-63I45 

110.958 

1.72627 

110.659 

1.82544 

110.392 

I.9292I 

110.153 

2.03784 

109.939 

I-2S 

1.64699 

116.026 

1.74096 

115-673 

1.83934 

115-356 

1.94237 

115-073 

2.05030 

114.818 

i-3 

1.66501 

120.991 

1.75802 

120.595 

1.85549 

120.240 

1.95767 

119.921 

2.06479 

119.634 

i-3S 

1.68497 

125.845 

1.77694 

125.418 

1.87343 

125-035 

1.97468 

124.689 

2.08093 

124.379 

1.4 

1.70635 

130.580 

1.79722 

130.135 

1.89268 

129-735 

1.99295 

129.373 

2.098^8 

129.047 

i  -45 

1.72856 

135-195 

1.81832 

134.745 

1.91273 

I34-338 

2.OI2OO 

I33.970 

2.11637 

133-638 

i-5 

I.75I04 

139-693 

1.83970 

139.248 

1.93306 

138.845 

2.03135 

138.480 

2.13478 

138.150 

i-SS 

1.77323 

144.077 

1.86084 

143.648 

1.95319 

143-258 

2.05051 

142.905 

2.15302 

142.584 

1.6 

1.79462 

148.354 

1.88123 

147.950 

1.97262 

147-583 

2.06903 

147.249 

2.17067 

146.945 

1.65 

I.8I470 

152.533 

1  .90040 

152.163 

1.99091 

151-825 

2.08647 

151-518 

2.18731 

151-238 

i-7 

1.83304 

156.625 

1.91792 

156.295 

2.00764 

155-994 

2.10244 

I55.7I9 

2.20254 

155469 

i-75 

1.84924 

160.640 

1.93341 

160.356 

2.02245 

160.097 

2.II658 

159.861 

2.21604 

I59.645 

1.8 

1.86297 

164.590 

1.94654 

164.358 

2.03500 

164.145 

2.12858 

163.951 

2.22751 

163.774 

1.85 

1.87394 

168.489 

1.95704 

168.311 

2.04505 

168.149 

2.13819 

168.000 

2.23669 

167.864 

1.9 

1.88193 

172.348 

1.96470 

172.228 

2.05238 

172.118 

2.14520 

172.018 

2-24334 

171.925 

I-9S 

1.88679 

176.180 

1.96935 

176.120 

2.05684 

176.065 

2.14947 

176.014 

2.24747 

175-968 

2.0 

1.88842 

180.000 

1.97091 

180.000 

2.05833 

180.000 

2.15090 

180.000 

2.24884 

180.000 

Example.      cosh  (1.35  -f  i  1.30)  =  1.85549  / 120°. 240  =  1.85549  / 120°.  14'.  24*. 


[in] 


TABLE  XI.    HYPERBOLIC   COSINES,     cosh  (x  +  ig)  =  r /y.     CONTINUED 


x  =  1.5 


=  1-55 


x  =  1.6 


*  =  1.65 


x  =  1.7 


o 

2.35241   o.ooo 

2.46186   o.ooo 

2.57746   o.ooo 

2.69952   o.ooo 

2.82832   o.ooo 

0.05 

2.35110   4.075 

2.46061   4.113 

2.57627  4.149 

2.69839   4.181 

2.82723   4.210 

o.r 

2.34720   8.158 

2.45688  8.235 

2.57272   8.305 

2.69499   8.369 

2.82399   8.427 

0.15 

2.34080  12.260 

2.45076  12.374 

2.56687  12.477 

2.68941  12.571 

2.81867   12.657 

O.2 

2.33202  16.389 

2.44238  16.536 

2.55887  16.671 

2.68178  16.794 

2.81138  16.906 

0.25 

2.32107  20.552 

2.43193  20.732 

2.54890  20.895 

2.67226   21.044 

2.80230  21.179 

o-3 

2.30819  24.759 

2.41976  24.967 

2.53717  25.155 

2.66108   25.327 

2.79164  25.483 

o-35 

2.29365   29.016 

2.40577  29.248 

2.52395  29.458 

2.64847  29.649 

2.77963   29.822 

0.4 

2-27779  33-33° 

2.39066  33.580 

2-50955  33-808 

2.63476  34.014 

2.76657  34.201 

0-45 

2.26098  37.706 

2.37465  37-970 

2.49430  38.209 

2.62024  38.426 

2.75274  38.622 

o-5 

2.24362  42.150 

2.35812  42.421 

2.47857  42.666 

2.60527  42.887 

2.73850  43.089 

0-55 

2.22612  46.663 

2.34148  46.934 

2.46274  47.180 

2.59021  47.401 

2.72418  47.602 

0.6 

2.20892   51-247 

2.32513  51.512 

2.44721  51-752 

2.57544  51.968 

2.71014  52.163 

0.65 

2.19245   55-901 

2.30949  56.154 

2.43235  56.381 

2.56133  56.585 

2.69673  56.771 

0.7 

2.17714  60.624 

2.29496  60.856 

2.41856  61.065 

2.54824  61.253 

2.68430  61.422 

o-7S 

2.16340  65.410 

2.28193  65.615 

2.40619  65.800 

2.53650  65.966 

2.67316  66.116 

0.8 

2.15159   70-254 

2.27074  70.426 

2-39558  70.581 

2.52644   70.720 

2.66361  70.845 

0.85 

2.14204  75-x45 

2.26169   75.279 

2.38701  75.400 

2.51831  75.508 

2-65591  75-605 

0.9 

2.13502  80.075 

2.25504  80.167 

2.38071   80.249 

2.51234  80.323 

2.65025  80.390 

o-9S 

2.13073  85.031 

2.25098  85.077 

2.37686  85.119 

.  2.50869  85.157 

2.64679  85.191 

I.O 

2.12928  90.000 

2.24961   90.000 

2-37557  90-000 

2.50747  90.000 

2.64563  90.000 

1.05 

2.13073  94-969 

2.25098  94.923 

2.37686  94.881 

2.50869  94-843 

2.64679  94.809 

i.i 

2.13502  99.925 

2.25504  99.833 

2.38071  99.751 

2.51234  99.677 

2.65025  99.610 

i-iS 

2.14204  104.855 

2.26169  104.721 

2.38701  104.600 

2.51831  104.492 

2.65591  104.395 

1.2 

2.15159  109.746 

2.27074  109.574 

2.39558  109.419 

2.52644  109.280 

2.66361  109.155 

I.2S 

2.16340  114-59° 

2.28193  114.385 

2.40619  114.200 

2.53650  114.034 

2.67316  113.884 

i~3 

2.17714  119.376 

2.29496  119.144 

2.41856  118.935 

2.54824  118.747 

2.68430  118.578 

•35 

2.19245  124.099 

2.30949  123.846 

2.43235  123.619 

2.56133  123.415 

2.69673  123.229 

•4 

2.20892  128.753 

2.32513  128.488 

2.44721  128.248 

2.57544  128.032 

2.71014  127.837 

•45 

2.22612  133.337 

2.34148  133.066 

2.46274  132.820 

2.59021  132.599 

2.72418  132.398 

•5 

2.24362  137-850 

2.35812  137.580 

2-47857  137-334 

2.60527  137.113 

2.73850  136.911 

•55 

2.26098  142.294 

2.37465  142.030 

2.49430  141-791 

2.62024  141.574 

2.75274  141.378 

.6 

2.27779  146.670 

2.39066  146.420 

2.50955  146.192 

2.63476  145.986 

2.76657  145.799 

•65 

2.29365  150.984 

2.40577  150.752 

2.52395  150.542 

2.64847  150.351 

2.77963  150.178 

•7 

2.30819  155-241 

2.41976  155.033 

2.537I7  I54.845 

2.66108  154.673 

2.79164  154.517 

•75 

2.32107  159.448 

2.43193  159.268 

2.54890  159.105 

2.67226  158.956 

2.80230  158.821 

.8 

2.33202  163.611 

2.44238  163.464 

2.55887  163.329 

2.68178  163.206 

2.81138  163.094 

•85 

2.34080  167.740 

2.45076  167.626 

2.56687  167.523 

2.68941  167.429 

2.81867  167.343 

•9 

2.34720  171.842 

2.45688  171.765 

2.57272  171.695 

2.69499  171.631 

2.82399  171.573 

•95 

2.35110  175-925 

2.46061  175.887 

2.57627  175.851 

2.69839  175.819 

2.82723  175.790 

2.0 

2.35241  180.000 

2.46186  180.000 

2.57746  180.000 

2.69952  180.000 

2.82832  180.000 

Example.       cosh  (1.6  +  i  1.6)  =  2.50955  /I46°.i92  =  2.50955  /i46°.n'.3iy. 


[112] 


TABLE  XI.    HYPERBOLIC   COSINES,     cosh  (*  +  iq)  =  r /y.     CONTINUED 


x  = 

1-75 

x  =  1.8 

x  = 

1.85 

*  = 

1.9 

X  = 

i-95 

9 

r 

y 

r 

Y 

r 

7 

r 

7 

r 

Y 

0 

0 

0 

o 

o 

0 

2.96419 

o.ooo 

3.10747 

O.OOO 

3. 

25853 

o.ooo 

3.4I773 

o.ooo 

3-58548 

o.ooo 

0.05 

2.96315 

4.237 

3.10648 

4.262 

3' 

25759 

4.284 

3.41683 

4-304 

3.58462 

4.322 

O.I 

2.96006 

8.480 

3-10353 

8.529 

3' 

25477 

8.572 

3-41415 

8.612 

3.58207 

8.648 

0.15 

2.95498 

12-735 

3.09869 

12.806 

3. 

25015 

12.871 

3-40975 

12.929 

3-57788 

12.983 

0.2 

2.94804 

17.008 

3.09206 

17.100 

3' 

24384 

17.184 

3-40373 

17.260 

3-572I3 

17.330 

0.25 

2.93938 

21.302 

3.08382 

21.414 

3' 

23598 

21.516 

3-39624 

21.608 

3-56500 

21.692 

O.3 

2.92921 

25-625 

3-074I3 

25-754 

3' 

22675 

25.871 

3-38745 

25-977 

3-55663 

26.073 

o-3S 

2.91777 

29.980 

3-06323 

30.123 

3' 

21636 

30.252 

3-37756 

30.369 

3-54721 

30.476 

0.4 

2.90532 

34-370 

3-05138 

34-524 

3' 

20507 

34.663 

3.36681 

34.789 

3-53698 

34.904 

0-45 

2.89217 

38.800 

3-03885 

38.961 

3' 

i93J5 

39.107 

3-35546 

39-239 

3.52617 

39.358 

o-5 

2.87861 

43.270 

3-02595 

43-435 

3 

,18088 

43.584 

3-34378 

43-719 

3-51507 

43.840 

0-55 

2.86499 

47.784 

3.01300 

47-947 

3' 

,16856 

48.096 

3-33207 

48.230 

3-50393 

48.351 

0.6 

2.85165 

52-339 

3.00031 

52.499 

3' 

15650 

52.643 

3.32060 

52.773 

3-49302 

52.890 

0.65 

2.83891 

56.937 

2.98821 

57.088 

3- 

14500 

57.224 

3-30967 

57.346 

3-48263 

57-457 

0.7 

2.82710 

6i.575 

2.97699 

61.713 

3. 

13434 

61.837 

3-29955 

61.949 

3-47301 

62.050 

0-75 

2.81653 

66.250 

2-96695 

66.371 

3' 

,12481 

66.480 

3.29049 

66.579 

3.46441 

66.668 

0.8 

2.80747 

70.958 

2-95835 

7L059 

3' 

,11665 

71.150 

3.28274 

71.233 

3-45702 

71-307 

0-85 

2.80016 

75.693 

2.95142 

75-772 

3' 

,11006 

75.842 

3.27649 

75.906 

3-45II2 

75-964 

0.9 

2.79479 

80.449 

2-94633 

80.503 

3 

10523 

80.552 

3.27191 

80.595 

3-44675 

80.635 

o-95 

2.79151 

85.221 

2.94322 

85.248 

3. 

,10228 

85-273 

3.26911 

85-295 

3.44410 

85-3I5 

I.O 

2.79041 

90.000 

2.94217 

90.000 

3- 

10129 

90.000 

3.26816 

90.000 

3-44321 

90.000 

1.05 

2.79151 

94-779 

2.94322 

94-752 

3 

,10228 

94.727 

3.26911 

94-705 

3.44410 

94.685 

i.i 

2-79479 

99-551 

2.94633 

99-497 

3, 

10523 

99.448 

3.27191 

99.405 

3-44675 

99-365 

1.15 

2.80016 

104.307 

2.95142 

104.228 

3' 

,11006 

104.158 

3-27649 

104.094 

3.45112 

104.036 

1.2 

2.80747 

109.042 

2-95835 

108.941 

3' 

,11665 

108.850 

3-28274 

108.767 

3-45702 

108.693 

1.25 

2.81653 

II3-750 

2.96695 

113.629 

3 

,12481 

113.520 

3.29049 

113.421 

3.46441 

II3-332 

2.82710 

118.425 

2.97699 

118.287 

3' 

13434 

118.163 

3-29955 

118.051 

3-47301 

117.950 

i-35 

2.83891 

123.063 

2.98821 

122.912 

3- 

,14500 

122.776 

3.30967 

122.654 

3.48263 

122.543 

2.85165 

127.661 

3.00031 

127.501 

3' 

15650 

127.357 

3.32060 

127.227 

3-49302 

127.110 

1-45 

2.86499 

132.216 

3.01300 

132-053 

3' 

16856 

131.904 

3-33207 

131.770 

3-50393 

131.649 

1-5 

2.87861 

136.730 

3-02595 

136-565 

3' 

,18088 

136.416 

3-34378 

136.281 

3-5I507 

136.160 

i-SS 

2.89217 

141.200 

3-03885 

141.039 

3' 

I93I5 

140.893 

3-35546 

140.761 

3-52617 

140.642 

1.6 

2.90532 

145.630 

3-05138 

I45-476 

3. 

20507 

145.337 

3.36681 

145.211 

3-53698 

145.096 

1-65 

2.91777 

150.020 

3-06323 

149.877 

3- 

21636 

149.748 

3-37756 

149.631 

3-54721 

149.524 

2.92921 

154-375 

3-07413 

154.246 

3' 

22675 

154.129 

3-38745 

154.023 

3-55663 

I53-927 

1-75 

2.93938 

158.698 

3.08382 

158-586 

3' 

23598 

158.484 

3-39624 

158.392 

3-56500 

158.308 

1.8 

2.94804 

162.992 

3.09206 

162.900 

3, 

24384 

162.816 

3.40373 

162.740 

3-572I3 

162.670 

1.85 

2.95498 

167.265 

3.09869 

167.194 

3. 

25015 

167.129 

3-40975 

167.071 

3-57788 

167.017 

1.9 

2.96006 

171.520 

3-10353 

171.471 

3' 

25477 

171.428 

3-41415 

171.388 

3-58207 

171-352 

i-95 

2.96315 

I75-763 

3.10648 

I75-738 

3' 

•25759 

175.716 

3.41683 

175-696 

3.58462 

175-678 

2.O 

2.96419 

180.000 

3.10747 

180.000 

3 

•25853 

180.000 

3-4I773 

180.000 

3.58548 

180.000 

Example. 

cosh  (1.95 

+  »'  1.25) 

=  3.4644I  /ii3°.33i 

!  =  3-46441  /H3°.I9'. 

s£ 

[113] 


TABLE  XI.    HYPERBOLIC   COSINES,     cosh  (*  +  iq)  =  r  /T.     CONTINUED 


x  =  2.05 


a;  =  2.15 


o 

0.05 

O.I 

0.15 

0.2 

3.76220   o.ooo 
.  3.76137   4-339 
3.75894   8.68! 
3-75495   13-031 
3.74948   17-392 

3.94832   o.ooo 

3.94753   4.354 
3.94521   8.711 
3.94142  13.075 
3.93620  17.449 

4.14431   o.ooo 

4-14357  4-368 
4.14136   8.738 

4-13774  13-115 
4.13278  17.501 

4.35067   o.ooo 

4.34996  4.380 
4.34785   8.763 
4.34440  13.151 
4.33968  17.548 

4.56791   o.ooo 

4.56723  4-391 
4.56523  8.785 
4.56167  13.184 

4.55744   I7-591 

0.25 

0-35 
0.4 

0-45 

3.74268  21.768 
3.73470  26.160 

3-72574  30-573 
3.71600  35.008 

3-70572  39467 

3.92972  21.836 
3.92213  26.239 
3-9r359  30-660 
3.90432  35.102 
3-89453  39-565 

4.12661   21.899 
4.11938  26.311 
4.11125  30.740 
4.10243  35.187 
4.09311  39.653 

4.33381  21.956 
4.32693  26.376 
4.31918  30.812 

4.31079  35-264 
4.30193  39.734 

4.55184  22.007 
4.54529  26.435 
4-53793  30-877 
4.52993   35-334 
4.52150  39.807 

0-5 
0-55 
0.6 
0.65 

o-7 

3.69515  43-951 
3.68455  48.461 
3.67418  52.996 
3.66430  57-557 
3.65517  62.142 

3.88448  44-051 
3.87440  48.560 
3-86454  53-092 

s-Sss^  57-647 

3.84648  62.224 

4.08355  44.141 
4.07396  48.649 
4.06459  53.179 
4.05566  57.729 

4.04740  62.299 

4.29282  44.223 

4.28371  48.730 
4.27479  53.257 
4.26630  57.803 
4.25846  62.366 

4.51285   44.297 
4.50417   48.804 
4-49570   53-329 
4.48763   57-870 
4.48017   62.427 

0-75 
0.8 
0.85 
0.9 
0-95 

•  3.64699  66.748 
3.64000  71.374 
3-63437   76-016 
3.63023  80.670 
3.62771   85.333 

3.83870  66.820 
3.83206  71-434 
3.82671   76.062 
3.82279  80.702 
3.82039  85.349 

4.04002  66.886 
4.03371   71.489 
4.02863   76.105 
4.02490  80.731 
4.02263  85.364 

4.25145  66.945 
4.24544  71.538 
4.24061  76-143 
4.23707  80.757 
4.23491  85.377 

4.47350  66.998 
4.46780   71.582 
4.46322   76.177 
4.45985   80.780 
4-45779  85.389 

I.O 

1.05 
i.i 

1.2 

3.62686  90.000 
3.62771   94.667 
3.63023  99.330 
3.63437  103.984 
3.64000  108.626 

3.81958  90.000 
3.82039  94-651 
3.82279  99.298 
3.82671  103.938 
3.83206  108.566 

4.02186  90.000 
4.02263  94-636 
4.02490  99.269 
4.02863  103.895 
4.03371  108.511 

4.23419  90.000 
4.23491  94.623 
4.23707  99.243 

4.24061  103.857 
4.24544  108.462 

4.45711  90.000 

4-45779   94-6ii 
4.45985   99.220 
4.46322  103.823 
4.46780  108.418 

1.25 
1.4 

i-45 

3.64699  113.252 
3.65517  117-858 
3.66430  122.443 
3.67418  127.004 
3.68455  I3I-539 

3.83870  113.180 
3.84648  117.776 
3.85515  122.353 
3.86454  126.908 
3.87440  131.440  « 

4.04002  113.114 
4.04740  117.701 
4.05566  122.271 
4.06459  126.821 
4.07396  131.351 

4.25145  113.055 
4.25846  117-634 

4.26630  122.197 

4.27479  126.743 
4.28371  131.270 

4.47350  113.002 
4.48017  117.573 
4.48763  122.130 
4.49570  126.671 
4.50417  131.196 

i-SS 
1.6 

1-65 

3.69515  136.049 
3.70572  140.533 
3.71600  I44-992 
3.72574  149-427 
3.73470  153.840 

3.88448  135-950 
3.89453  140.435 
3.90432  144.898 
3-9I359  149-340 
3.92213  153.761 

4.08355  135.859 
4.09311  140.347 
4.10243  144.813 
4.11125  149.260 
4.11938  153.689 

4.29282  135.778 
4.30193  140.266 
4.31079  144-736 

4.31918  149.188 

4.32693  153.624 

4.51285  135.703 
4.52150  140.193 
4.52993  144.666 
4-53793  149-123 
4.54529  153.565 

1.8 
1.85 
1.9 

1-95 

3.74268  158.232 
3.74948  162.608 
3.75495  166.969 
3.75894  171-319 
3.76137  i75-66i 

3.92972  158.164 
3.93620  162.551 
3.94142  166.925 
3.94521  171.289 
3-94753  I75-646 

4.12661  158.101 
4.13278  162.499 
4.13774  166.885 
4.14136  171.262 
4.14357  175.632 

4.33381  158.044 
4.33968  162.452 
4.34440  166.849 
4.34785  171.237 
4.34996  175-620 

4.55184  157-993 
4.55744  162.409 
4.56167  166.816 
4.56523  171.215 
4.56723  175.609 

2.0 

3.76220  180.000 

3.94832  180.000 

4.14431  180.000 

4-35067  180.000 

4.56791  180.000 

Example,      cosh  (2.0  +  i  0.5)  =  3.69515 


=  3-695*5  /43°-57'.04g- 


C"4] 


TABLE  XI.    HYPERBOLIC  COSINES,     cosh  (x  +  iq)  =  7/7.    CONTINUED 


X  = 

2.25 

x  = 

2-3 

x  = 

2-35 

x  = 

2.4 

T  ST 

2-45 

q 

r 

y 

r 

y 

r 

y 

r 

y 

r 

y 

o 

o 

0 

0 

o 

0 

4-79657 

O.OOO 

5.03722 

O.OOO 

5- 

29047 

O.OOO 

5-55695 

O.OOO 

S-83732 

0.000 

0.05 

4-79593 

4.401 

5.03661 

4411 

5- 

28989 

4419 

5-55639 

4.427 

5.83680 

4-434 

O.I 

4.79402 

8.805 

5A3479 

8.824 

5- 

28816 

8.840 

5-55474 

8.856 

5-83522 

8.869 

0.15 

4.79088 

13.214 

5.03181 

13.241 

5. 

28531 

I3-265 

5-55204 

13.288 

5-83265 

13-308 

O.2 

4.78661 

17.629 

5-02773 

17.664 

5- 

28143 

17.696 

5-54834 

17-725 

5.82913 

I7.75I 

0.25 

4.78127 

22.053 

5.02265 

22.096 

5' 

,27662 

22.134 

5-54375 

22.169 

5.82476 

22.200 

°-3 

4-77503 

26.488 

5.01672 

26-537 

5' 

,27096 

26.581 

5-53837 

26.620 

5.81964 

26.656 

o-3S 

4.76802 

30.936 

5.01005 

30.989 

5' 

,26461 

3I-038 

5-53233 

31.081 

5-81389 

31.121 

0.4 

4.76042 

35-397 

5.00281 

35-454 

5- 

25772 

35-506 

5-52578 

35-553 

5.80765 

35-595 

0-45 

4.75240 

39-873 

4.99518 

39-932 

5- 

25046 

39.986 

5-51887 

40-035 

5.80108 

40.079 

o-5 

4-744I5 

44-364 

4-98734 

44.424 

5' 

23728 

44-479 

5-5II77 

44-529 

5-79434 

44-573 

o-55 

4-73592 

48.870 

4-97950 

48.930 

5' 

23553 

48.984 

5.50468 

49.034 

5.78758 

49.078 

0.6 

4.72784 

53-392 

4.97183 

53-451 

5 

,22825 

53-503 

5-49775 

53-551 

5-78099 

53-593 

0.65 

4.72017 

57-930 

4.96454 

57-984 

5' 

22131 

58-034 

5-49II5 

58.078 

5-77472 

58.119 

0.7 

4.71308 

62.482 

4.95780 

62.531 

5' 

,21490 

62.576 

5-48505 

62.617 

5.76892 

62.653 

0-75 

4.70675 

67.046 

4-95I77 

67.090 

5 

.20917 

67.129 

5.47961 

67.165 

5-76375 

67.197 

0.8 

4.70134 

71.622 

4.94662 

7I-659 

5 

.20428 

71.691 

5-47495 

71.721 

5-75932 

71.748 

0.85 

4.69697 

76.208 

4.94249 

76-236 

5 

.20035 

76.261 

5.47122 

76.284 

5^5576 

76-305 

0.9 

4-69377 

80.801 

4-93944 

80.820 

5 

.19745 

80.837 

5.46847 

80.853 

5-753i6 

80.867 

o-9S 

4.69182 

85-399 

4-93759 

85.409 

5 

.19569 

85.418 

5.46679 

85.426 

5.75I56 

85433 

I.O 

4.69117 

90.000 

4.93696 

90.000 

5 

.19510 

90.000 

5.46623 

90.000 

5-75103 

90.000 

1.05 

4.69182 

94.601 

4-93759 

94.591 

5 

.19569 

94.582 

5.46679 

94-574 

5.75I56 

94-567 

i.i 

4-69377 

99.199 

4-93944 

99.180 

5 

.19745 

99.163 

5.46847 

99.147 

5.753i6 

99-133 

i-iS 

4.69697 

103.792 

4.94249 

103.764 

5 

-20035 

103-739 

5.47122 

103.716 

5^5576 

103.695 

1.2 

4-70I34 

108.378 

4.94662 

108.341 

5 

.20428 

108.309 

5-47495 

108.279 

5-75932 

108.252 

1-25 

4.70675 

112.954 

4-95I77 

112.910 

5 

.20917 

112.871 

5.47961 

112.835 

5.76375 

112.803 

J-3 

4.71308 

117.518 

4-9578o 

117.469 

5 

.21490 

117.424 

5-48505 

117-383 

5.76892 

"7-347 

I-3S 

4.72017 

122.070 

4.96454 

122.016 

5 

.22131 

121.966 

5-49"5 

121.922 

5-77472 

121.881 

1.4 

4.72784 

126.608 

4.97183 

126.549 

5 

.22825 

126.497 

5-49775 

126.449 

5.78099 

126.407 

i-4S 

4-73592 

131.130 

4-97950 

131.070 

5 

•23553 

131.016 

5.50468 

130.966 

5.78758 

130.922 

i-5 

4-744IS 

I35-636 

4-98734 

I35-576 

5 

.23728 

I35-52I 

5-5II77 

I35-47I 

5-79434 

135427 

i-SS 

4.75240 

140.127 

4.99518 

140.068 

5 

.25046 

140.014 

5-51887 

I39-965 

5.80108 

139.921 

1.6 

4.76042 

144.603 

5.00281 

144.546 

5 

.25772 

144.494 

5-52578 

144.447 

5-80765 

144.405 

1-65 

4.76802 

149.064 

5.01005 

149.011 

5 

.26461 

148.962 

5-53233 

148.919 

5-81389 

148.879 

i-7 

4-77503 

I53-5I2 

5.01672 

I53-463 

5 

.27096 

I53-4I9 

5-53837 

i53-38o 

5.81964 

153-344 

i-7S 

4.78127 

157-947 

5.02265 

157.904 

5 

.27662 

157-866 

5-54375 

157-831 

5.82476 

157.800 

1.8 

4.78661 

162.371 

5-02773 

162.336 

5 

.28143 

162.304 

5-54834 

162.275 

5-82913 

162.249 

1.85 

4.79088 

166.786 

5.03181 

166.759 

5 

.28531 

166.735 

5-55204 

166.712 

5-83265 

166.692 

1.9 

4.79402 

I7I-I95 

5-03479 

171.176 

5 

.28816 

171.160 

5-55474 

171.144 

5-83522 

171.131 

i-95 

4-79593 

175-599 

5.03661 

I75-589 

5 

.28989 

I75-58I 

5-55639 

175-573 

5-83680 

I75.566 

2.0 

4-79657 

180.000 

5.03722 

180.000 

5 

.29047 

180.000 

5.55695 

180.000 

5.83732 

180.000 

Example.      cosh  (2.40  +  i  2.0)  =  5.55695  /i8o°  =  5.55695  \i8o°. 


["Si 


TABLE  XL    HYPERBOLIC  COSINES,     cosh  (x  +  iq)  =  r  /T.    CONTINUED 


*  =  2.5 

a  =  2.55 

*  =  2.6 

x  =  2.65 

*  =  2.7 

9 

r     7 

r     V 

r     7 

r     7 

r      V 

0 

o 

o 

0 

e 

o 

6.13229   o.ooo 

6.44259   o.ooo 

6.76901   o.ooo 

7.11234   o.ooo 

7-47347   o.ooo 

0.05 

6.13179   4-440 

6.44212   4.446 

6.76855  4.451 

7.11191  4.456 

7.47306   4.460 

O.I 

6.13030   8.881 

6.44069   8.893 

6.76720   8.903 

7.11062   8.912 

7.47183   8.920 

0.15 

6.12785   13.326 

6.43836  13.342 

6.76499  13.357 

7.10851  13.371 

7.46982   13.383 

O.2 

6.12450  17-774 

6.43518  17-796 

6.76195  17.815 

7.10563  17.833 

7.46708   17.848 

0.2S 

6.12034  22.228 

6.43121  22.254 

6.75818  22.278 

7.10204  22.299 

7.46366  22.318 

°-3 

6.11547   26.689 

6.42658  26.718 

6.75376  26.745 

7.09784  26.770 

7.45966  26.791 

o-35 

6.10999  31.157 

6.42137  31.190 

6.74881  31.219 

7.09313  31.246 

7-455i8  31.270 

0.4 

6.10406  35-634 

6.41572  35.668 

6-74343   35-700 

7.08802  35.728 

7.45032  35.754 

0-45 

6.09781  40.119 

6.40977  40.155 

6.73778  40.188 

7.08264  40.218 

7.44519  40.245 

o-5 

6.09139  44.614 

6.40367  44.651 

6.73197   44-684 

7.07711  44-713 

7-43994  44-741 

0-55 

6.08496  49.118 

6.39755  49-155 

6.72616  49.187 

7.07158  49-217 

7.43468  49.244 

0.6 

6.07869  53.632 

6.39160  53-667 

6.72048   53-699 

7.06618  53.727 

742955  53-754 

0.65 

6.07273  58.155 

6.38592  58.188 

6.71509   58.218 

7.06105  58.244 

7.42467  58.269 

0.7 

6.06722  62.686 

6.38068  62.716 

6.71011   62.743 

7.05631  62.768 

7.42017  62.790 

0-75 

6.06229  67.226 

6.37601  67.252 

6.70565   67.275 

7.05208  67.297 

7.41614  67.316 

0.8 

6.05808  71.772 

6.37201  71-794 

6.70185   71.813 

7.04847  71.831 

7.41271   71.847 

0.85 

6.05471   76-324 

6.36879  76.340 

6.69880   76.356 

7.04556  76.369 

7.40994  76.382 

0.9 

6.05223  80.880 

6.36644  80.891 

6.69656   80.902 

7.04343  80.911 

7.40791   80.920 

o-95 

6.05071   85.439 

6.36499  85.445 

6.69518   85.450 

7.04213  85.455 

7.40668  85.460 

I.O 

6.05020  90.000 

6.36451  90.000 

6.69473   90.000 

7.04169  90.000 

7.40626  90.000 

1.05 

6.05071  94-561 

6.36499  94-555 

6.69518  94-55° 

7.04213  94.545 

7.40668  94.540 

i.i 

6.05223  99.120 

6.36644  99.109 

6.69656   99.098 

7.04343  99.089 

7.40791   99.080 

i-iS 

6.05471  103.676 

6.36879  103.660 

6.69880  103.644 

7.04556  103.631 

7.40994  103.618 

1.2 

6.05808  108.228 

6.37201  108.206 

6.70185  108.187 

7.04847  108.169 

7.41271  108.153 

1.25 

6.06229  112.774 

6.37601  112.748 

6.70565  112.725 

7.05208  112.703 

7.41614  112.684 

i-3 

6.06722  117.314 

6.38068  117.284 

6.71011  117.257 

7.05631  117.232 

7.42017  117.210 

i-3S 

6.07273  121.845 

6.38592  121.812 

6.71509  121.782 

7.06105  121.756 

7.42467  121.731 

1.4 

6.07869  126.368 

6.39160  126.333 

6.72048  126.301 

7.06618  126.273 

7.42955  126.246 

i-45 

6.08496  130.882 

6-39755  130-845 

6.72616  130.813 

7.07158  130.783 

7.43468  130.756 

i-5 

6.09139  135.386 

6.40367  135-35° 

6-73J97  i35-3i6 

7.07711  135.286 

7-43994  I35-259 

i-55 

6.09781  139.881 

6.40977  139-845 

6.73778  139.812 

7.08264  139.782 

7-445I9  139-755 

1X3 

6.10406  144.366 

6.41572  144-332 

6.74343  144.300 

7.08802  144.272 

7.45032  144.246 

1.65 

6.10999  148.843 

6.42137  148.810 

6.74881  148.781 

7.09313  148.754 

7.45518  148.730 

i-7 

6.11547  153.3" 

6.42658  153.282 

6.75376  I53-255 

7.09784  153.230 

7.45966  153.209 

i-75 

6.12034  157.772 

6.43121  157-746 

6.75818  157-722 

7.10204  157.701 

7.46366  157.682 

1.8 

6.12450  162.226 

6.43518  162.204 

6.76195  162.185 

7.10563  162.167 

7.46708  162.152 

1.85 

6.12785  166.674 

6.43836  166.658 

6.76499  166.643 

7.10851  166.629 

7.46982  166.617 

i.  g 

6.13030  171.119 

6.44069  171.107 

6.76720  171.097 

7.11062  171.088 

7.47183  171.080 

1-95 

6.13179  175-560 

6.44212  175.554 

6.76855  175-549 

7.11191  175.544 

7.47306  175.540 

2.0 

6.13229  180.000 

6.44259  180.000 

6.76901  180.000 

7.11234  180.000 

7-47347  180.000 

Example. 

cosh  (2.65  +  i  0.75) 

=  7.05208  /67°.297 

=  7.05208  /67°.i7' 

.49'. 

TABLE  XI.     HYPERBOLIC   COSINES,     cosh  (x  +  iq)  =  r  /T.     CONTINUED 


x  =  2.75 


x  =.2.85 


*  =  2.9 


x  =  2.95 


9 

r 

7 

r 

7 

r 

7 

r 

7 

r 

7 

o 

o 

o 

o 

o 

o 

7-85328 

O.OOO 

8.25273 

O.OOO 

8.67281 

O.OOO 

9.11458 

O.OOO 

9.57915 

O.OOO 

0.05 

7.85288 

4.464 

8-25235 

4.467 

8.67246 

4.470 

9.11424 

4-473 

9-57882 

4.476 

O.I 

7.85172 

8.928 

8.25124 

8-935 

8.67140 

8.941 

9.11324 

8-947 

9.57787 

8.952 

0.15 

7.84980 

13-394 

8.24942 

13.404 

8.66967 

13.413 

9.11160 

13.421 

9-57630 

13-429 

0.2 

7.84720 

17.863 

8.24694 

17.876 

8.66731 

17.888 

9.10934 

17.898 

9.57416 

17.908 

0.25 

7-84395 

22-335 

8.24385 

22.351 

8.66436 

22.364 

9.10655 

22.378 

9-57150 

22.389 

o-3 

7.84015 

26.8II 

8.24024 

26.829 

8.66092 

26.845 

9.10327 

26.860 

9.56838 

26.873 

035 

7.83588 

31.292 

8.23617 

3I.3I2 

8.65706 

3J.330 

9.09959 

3I-346 

9.56488 

31-360 

0.4 

7-83125 

35-778 

8.23177 

35-799 

8.65287 

35-8i8 

9.09561 

35-83S 

9.56109 

35-851 

o-4S 

7.82638 

40.269 

8.22713 

40.291 

8.64846 

40.311 

9.09142 

40-329 

9-557II 

40.345 

o-S 

7.82138 

44.766 

8.22238 

44-788 

8.64395 

44.808 

9.08711 

44.827 

9-55301 

44.843 

o-55 

7.81637 

49.269 

8.21762 

49.290 

8.63941 

49.310 

9.08281 

49-329 

9.54892 

49-345 

0.6 

7.81149 

53-777 

8.21298 

53-798 

8.63500 

53-817 

9.07861 

53-835 

9.54492 

53-850 

0.65 

7.80685 

58.291 

8.20857 

58-311 

8.63080 

58-329 

9.07461 

58.345 

9-54113 

58-360 

0.7 

7.80257 

62.810 

8.20449 

62.828 

8.62691 

62.845 

9.07093 

62.859 

9.53762 

62.873 

0-75 

7-79875 

67-334 

8.20085 

67-350 

8.62347 

67.364 

9.06764 

67-377 

9-53450 

'  67-389 

0.8 

7-79547 

71.862 

8.19774 

7I-875 

8.62051 

71.887 

9.06483 

71.898 

9-53182 

71.907 

0.85 

7-79285 

76.393 

8.19524 

76.404 

8.61813 

76.413 

9-06257 

76.421 

9.52967 

76.429 

0.9 

7.79092 

80.927 

8.19341 

80.934 

8.61639 

80.940 

9.06091 

80.946 

9.52809 

80.951 

o-95 

7-78975 

85-463 

8.19230 

85.467 

8.61532 

85-470 

9.05990 

85-473 

9-52713 

85475 

I.O 

7-78935 

90.000 

8.19192 

90.000 

8.61497 

90.000 

9.05956 

90.000 

9.52681 

90.000 

1.05 

7-78975 

94-537 

8.19230 

94-533 

8.61532 

94-530 

9.05990 

94.527 

9-52713 

94.525 

i.i 

7.79092 

99.073 

8.19341 

99.066 

8.61639 

99.060 

9.06091 

99.054 

9.52809 

99.049 

i-iS 

7.79285 

103.607 

8.19524 

103.596 

8.61813 

103.587 

9.06257 

103.579 

9.52967 

103.571 

1.2 

7-79547 

108.138 

8.19774 

108.125 

8.62051 

108.113 

9.06483 

108.102 

9-53182 

108.093 

".«S 

7-79875 

112.666 

8.20085 

112.650 

8.62347 

112.636 

9.06764 

112.623 

9-53450 

112.611 

i-3 

7.80257 

117.190 

8.20449 

117.172 

8.62691 

"7-155 

9.07093 

117.141 

9-53762 

117.127 

•35 

7.80685 

121.709 

8.20857 

121.689 

8.63080 

121.671 

9.07461 

121.655 

9.54ii3 

121.640 

•4 

7.81149 

126.223 

8.21298 

126.202 

8.63500 

126.183 

9.07861 

126.165 

9-54492 

126.150 

•45 

7.81637 

I30-73I 

8.21762 

130.710 

8.63941 

130.690 

9.08281 

130.671 

9.54892 

130.655 

•5 

7-82138 

I35-234 

8.22238 

135.212 

8.64395 

I35-I92 

9.08711 

I35.I73 

9-55301 

135.157 

•55 

7.82638 

I39-73I 

8.22713 

139.709 

8.64846 

139.689 

9.09142 

139.671 

9-557II 

I39.655 

.6 

7-83125 

144.222 

8-23177 

144.201 

8.65287 

144.182 

9.09561 

144.165 

9.56109 

144.149 

•6S 

7-83588 

148.708 

8.23617 

148.688 

8.65706 

148.670 

9.09959 

148.654 

9-56488 

148.640 

•7 

7.84015 

153-189 

8.24024 

i53-I7i 

8.66092 

I53-I55 

9.10327 

153-140 

9.56838 

I53""? 

•75 

7.84395 

I57-665 

8.24385 

157.649 

8.66436 

I57-636 

9.10655 

157.622 

9-57I50 

157-611 

.8 

7.84720 

162.137 

8.24694 

162.124 

8.66731 

162.112 

9.10934 

162.102 

9.57416 

162.092 

•85 

7.84980 

166.606 

8.24942 

166.596 

8.66967 

166.587 

9.11160 

166.579 

9-57630 

166.571 

•9 

7.85172 

171.072 

8.25124 

171.065 

8.67140 

171.059 

9.11324 

171-053 

9.57787 

171.048 

•95 

7.85288 

I75-536 

8-25235 

175.533 

8.67246 

175-530 

9.11424 

I75.527 

9-57882 

I75-524 

2.O 

7-85328 

180.000 

8.25273 

180.000 

8.67281 

180.000 

9.11458 

180.000 

9.579I5 

180.000 

Example. 

cosh  (2.90  +  i  0.9) 

=  9.06091  /8o°.946 

=  0.96091  /8o°.<;6'.46'. 

[117] 


TABLE  XL    HYPERBOLIC  COSINES,     cosh  (x  +  iq)  =  r  /y.    CONTINUED 


x  = 

3-o 

x  =  3.05 

*  =  3-i 

x  =  3.15         x  = 

3-2 

<7 

r 

T 

r 

7 

r 

y 

r 

7 

r 

•y 

0 

0 

0 

o 

o 

0 

10.06766 

o.ooo 

10.58135 

O.OOO 

11.12150 

O.OOO 

11.68946 

o.ooo 

12.28665 

O.OOO 

0.05 

10.06737 

4.478 

10.58110 

4.480 

II.I2I2O 

4.482 

11.68920 

4.484 

12.28640 

4-485 

O.I 

10.06645 

8.956 

10.58020 

8.960 

II.I2O4O 

8.964 

11.68840 

8.968 

12.28560 

8.971 

0.15 

10.06500 

13-436 

10.57880 

13.442 

II.II9OO 

13-447 

11.68710 

13-452 

12.28440 

13.457 

O.2 

10.06292 

17.917 

10.57680 

I7-925 

II.II72O 

17.932 

11.68540 

17.938 

12.28280 

17.944 

0.25 

10.06040 

22.400 

10.57440 

22.409 

II.II49O 

22.418 

11.68320 

22.426 

12.28070 

22.433 

o-3 

10.05743 

26.886 

10.57160 

26.896 

II.II22O 

26.906 

11.68060 

26.915 

12.27820 

26.923  . 

0-35 

10.05410 

3J-374 

10.56840 

3I-386 

II.IO92O 

31.396 

11.67780 

31.406 

12.27550 

3!-4i5 

0.4 

10.05050 

35-865 

10.56500 

35.878 

II.IO6OO 

35.890 

11.67470 

35-900 

12.27260 

35.910 

0-45 

10.04670 

40.360 

10.56140 

40-373 

II.IO25O 

40.385 

11.67140 

40.396 

12.26950 

40.406 

o-S 

10.04280 

44.858 

10-55770 

44.871 

II.O99OO 

44.884 

11.66810 

44.895 

12.26630 

44.905 

10.03890 

49.360 

10.55400 

49-373 

11.09540 

49.385 

11.66470 

49.396 

12.26310 

49.406 

0.6 

10.03510 

53-865 

10.55040 

53-878 

II.0920O 

53-889 

11.66140 

53.900 

12.26000 

53-909 

0.65 

10.03150 

58.373 

10.54690 

58-385 

II.08880 

58.396 

11.65830 

58.406 

12.25700 

58-415 

0.7 

10.02820 

62.885 

10.54380 

62.896 

11.08580 

62.906 

11.65540 

62.915 

12.25430 

62.923 

0-75 

10.62520 

67-399 

10.54090 

67.409 

II.O83IO 

67.417 

11.65290 

67-425 

12.25180 

67.432 

0.8 

10.02260 

71.916 

10.53850 

71.924 

II.o8o8o 

71.932 

11.65070 

71-938 

12.24980 

71.944 

0.85 

10.02060 

76.435 

10.53660 

76.441 

11.07890 

76.447 

11.64890 

76.452 

12.24810 

76.457 

0.9 

10.01910 

80.956 

10.53520 

80.960 

11.07750 

80.964 

11.64770 

80.967 

12.24690 

80.970 

o-95 

10.01820 

85.478 

10.53430 

85.480 

II.O767O 

85.482 

11.64690 

85.484 

12.24610 

85-485 

I.O 

10.01787 

90.000 

10-53399 

90.000 

11.07645 

9O.OOO 

11.64661 

90.000 

12.24588 

90.000 

1.05 

10.01820 

94.522 

10.53430 

94.520 

II.O767O 

94.518 

11.64690 

94.516 

12.24610 

94.5I5 

i.i 

10.01910 

99.044 

10.53520 

99.040 

11.07750 

99.036 

11.64770 

99-033 

12.24690 

99.030 

I-I5 

10.02060 

103-565 

10.53660 

103.559 

11.07890 

103-553 

11.64890 

103.548 

12.24810 

103-543 

1.2 

10.02260 

108.084 

10.53850 

108.076 

II.o8o8o 

108.068 

11.65070 

108.062 

12.24980 

108,056 

!.25 

10.02520 

112.601 

10.54090 

112.591 

II.O83IO 

112.583 

11.65290 

112.575 

12.25180 

112.568 

1-3 

10.02820 

117.115 

10.54380 

117.104 

11.08580 

117.094 

11.65540 

117.085 

12.25430 

117.077 

i-35 

10.03150 

121.627 

10.54690 

121.615 

II.08880 

I2I.6O4 

11.65830 

121.594 

12.25700 

121.585 

1.4 

10.03510 

126.135 

10.55040 

126.122 

II.O92OO 

I26.III 

11.66140 

126.100 

12.26000 

126.091 

i-45 

10.03890 

130.640 

10.55400 

130.627 

11.09540 

130.615 

11.66470 

130.604 

12.26310 

130.594 

i-5 

10.04280 

I35.I42 

10.55770 

I35-I29 

II.O9900 

I35.II6 

11.66810 

135-105 

12.26630 

I35.095 

i-55 

10.04670 

139.640 

10.56140 

139.627 

II.IO250 

139.615 

11.67140 

139.604 

12.26950 

139-594 

1.6 

10.05050 

I44.I35 

10.56500 

144.122 

II.IO6OO 

I44.IIO 

11.67470 

144.100 

12.27260 

144.090 

1-65 

10.05410 

148.626 

10.56840 

148.614 

11.10920 

148.604 

11.67780 

148.594 

12.27550 

148.585 

10.05743 

I53-II4 

10.57160 

I53-I04 

II.II22O 

I53-094 

11.68060 

153-085 

12.27820 

I53-077 

i-75 

10.06040 

157.600 

10.57440 

I57-59I 

II.II490 

I57-582 

11.68320 

157-574 

12.28070 

I57-567 

1.8 

10.06292 

162.083 

10.57680 

162.075 

II.II72O 

162.068 

11.68540 

162.062 

12.28280 

162.056 

1-85 

10.06500 

166.564 

10.57880 

166.558 

n.II90O 

166.553 

11.68710 

166.548 

12.28440 

166.543 

1.9 

10.06645 

171.044 

10.58020 

171.040 

11.12040 

171.036 

11.68840 

171.032 

12.28560 

171.029 

I-9S 

10.06737 

I75-522 

10.58110 

I75-520 

II.I2I20 

I75-5I8 

11.68920 

175-516 

12.28640 

I75-5I5 

2.0 

10.06766 

180.000 

10.58135 

180.000 

II.I2I50 

iSo.OOO 

11.68946 

180.000 

12.28665 

iSo.OOO 

Example. 

cosh  (3.15 

+  2Q.I5) 

=  11.68710 

/i3°.452  =  11.68710  /i3°.27'.07*. 

TABLE  XI.    HYPERBOLIC   COSINES,     cosh  (*  +  iq)  =  r  £y.    CONTINUED 


=  3-25 


*' =  3-3 


=  3-35 


x  =  3.4 


=  3-45 


9 

r 

7 

r 

V 

r 

7 

r 

7 

r 

y 

o 

o 

o 

o 

0 

o 

12.91456 

0.000 

I3-57476 

O.OOO 

14.26891 

O.OOO 

14.99874 

o.ooo 

15.76607 

O.OOO 

0-05 

12.91430 

4.486 

13-57455 

4.488 

14.26870 

4.489 

4.490 

15-76587 

4491 

O.I 

12.91360 

8-973 

I3-57387 

8.976 

14.26805 

8.978 

14.99790 

8.980 

15-76530 

8.982 

0.15 

12.91240 

13.461 

I3-57275 

13-465 

14.26700 

13.468 

14.99692 

13-471 

iS-76433 

13-474 

O.2 

12.91085 

17.949 

i3-57!23 

17.954 

14.26556 

17-959 

14-99555 

17.963 

15-76303 

17.966 

0.25 

12.90888 

22.439 

13-56936 

22.445 

14.26377 

22.450 

14.99385 

22.455 

15.76144 

22.459 

o-3 

12.90658 

26.930 

13.56718 

26.937 

14.26167 

26.943 

14.99186 

26.948 

I5-75950 

26.953 

o-35 

12.90398 

31.423 

13.56470 

31.431 

14.25933 

31-437 

14.98960 

3L443 

I5-75740 

31-449 

0.4 

12.90117 

35-9x8 

13.56203 

35-926 

14.25614 

35-933 

14.98721 

35-939 

i5-755io 

35-945 

o-45 

12.89820 

40.415 

I3-559I8 

40.423 

14.25412 

40.430 

14.98466 

40-437 

15.75270 

40.443 

0-5 

12.89518 

44.914 

I3-55633 

44.922 

14-25137 

44-929 

14.98205 

44.936 

15.75020 

44.942 

0-55 

12.89215 

49-4I5 

I3-55346 

49.423 

14.24863 

49.430 

14-97945 

49-437 

I5-74772 

49-443 

0.6 

12.88910 

53.918 

13.55062 

53.926 

14.24595 

53-933 

14.97690 

53-939 

I5-74529 

53-945 

0.65 

12.88637 

58-423 

13.54796 

58.431 

14.24339 

58.437 

14.97448 

58.443 

15.74300 

58.449 

o-7 

12.88380 

62.930 

I3-54550 

62.937 

14.24106 

62.943 

14.97225 

62.948 

15.74088 

62.953 

0-75 

12.88146 

67439 

I3-54322 

67.445 

14.23897 

67450 

14.97023 

67455 

15-73898 

67459 

0.8 

12.87949 

71.949 

I3-54I4I 

71-954 

14.23718 

71-959 

14.96855 

71.963 

I5-73736 

71.966 

0.85 

12.87790 

76.461 

I3-5399I 

76.465 

I4-23573 

76.468 

14.96716 

76.471 

15-73603 

76.474 

0.9 

12.87670 

80.973 

13-53878 

80.976 

14.23470 

80.978 

14.96616 

80.980 

i5-735io 

80.982 

0-95 

12.87600 

85.486 

13-53810 

85.488 

14.23403 

85-489 

14.96556 

85.490 

I5-73450 

85.491 

I.O 

12.87578 

90.000 

13.53788 

90.000 

14.23382 

90.000 

14.96536 

90.000 

I5-73432 

90.000 

1.05 

12.87600 

94-5H 

13-53810 

94.512 

14.23403 

94-5" 

14.96556 

94.510 

I5-73450 

94.509 

i.i 

12.87670 

99.027 

I3-53878 

99.024 

14.23470 

99.022 

14.96616 

99.020 

i5-735io 

99.018 

1.15 

12.87790 

103.539 

I3-5399I 

103.535 

I4-23573 

103.532 

14.96716 

103.529 

15-73603 

103.526 

1.2 

12.87949 

108.051 

I3-54I4I 

108.046 

14.23718 

108.041 

14.96855 

108.037 

I5-73736 

108.034 

T.25 

12.88146 

112.561 

I3-54322 

"2.555 

14.23897 

"2.550 

14.97023 

"2-545 

15-73898 

112.541 

1.3 

12.88380 

117.070 

I3.54550 

117.063 

14.24106 

117.057 

14.97225 

117.052 

15.74088 

117.047 

1.35 

12.88637 

121.577 

I3-54796 

121.569 

14.24339 

121.563 

14.97448 

121-557 

15.74300 

121-551 

1.4 

12.88910 

126.082 

13.55062 

126.074 

14.24595 

126.067 

14.97690 

126.061 

I5-74529 

126.055 

i-45 

12.89215 

130-585 

I3-55346 

130.577 

14.24863 

130.570 

14.97945 

130.563 

I5-74772 

130.557 

i-S 

12.89518 

135.086 

I3-55633 

135-078 

14-25137 

I35-07I 

14.98205 

135.064 

15.75020 

I35-058 

i-55 

12.89820 

139-585 

I3-559I8 

139-577 

14.25412 

I39-570 

14.98466 

139-563 

15-75270 

139-557 

1.6 

12.90117 

144.082 

13.56203 

144.074 

14.25614 

144.067 

14.98721 

144.061 

i5-755io 

I44.055 

1.65 

12.90398 

148.577 

13.56470 

148.569 

14-25933 

148.563 

14.98960 

148.557 

I5-75740 

148.551 

12.90658 

I53-070 

13.56718 

153-063 

14.26167 

I53.057 

14.99186 

153-052 

I5-75950 

I53-047 

1-75 

12.90888 

I57-56I 

13-56936 

157-555 

14.26377 

IS7.550 

14-99385 

157-545 

15.76144 

I57.54I 

1.8 

12.91085 

162.051 

13-57123 

162.046 

14.26556 

162.041 

14-99555 

162.037 

15-76303 

162.034 

1.85 

12.91240 

166.539 

I3-57275 

166.535 

14.26700 

166.532 

14.99692 

166.529 

I5-76433 

166.526 

1.9 

12.91360 

171.027 

I3-57387 

171.024 

14.26805 

171.022 

14.99790 

171.020 

15-76530 

171.018 

i-95 

12.91430 

I75-5I4 

13-57455 

175-512 

14.26870 

175-5" 

14.99853 

i75-5io 

15-76587 

I75.509 

2.0 

12.91456 

180.000 

I3-57476 

180.000 

14.26891 

180.000 

14.99874 

180.000 

15.76607 

180.000 

Example.       cosh  (3.4  +  to. 75)  =  14.97023 


=  14-97023  /67°.27'.i8». 


["9] 


TABLE  XI.    HYPERBOLIC  COSINES,     cosh  (x  +  iq)  =  r  /y.    CONTINUED 


x  =  3-5 

x  =  3-55 

x  =  3.6 

x  =  3-65 

*  =  3-7 

9 

r      y 

r      y 

r      y 

r      y 

r      y 

o 

o 

o 

0 

0 

o 

16.57282   o.ooo 

17.42102   o.ooo 

18.31278   o.ooo 

19.25033   o.ooo 

20.23601   o.ooo 

0.05 

16.57260   4.492 

17.42083   4.493 

18.31260   4.493 

19.25015   4.494 

20.23585  4.495 

O.I 

16.57210   8.984 

17.42030   8.985 

18.31210   8.987 

19.24970   8.988 

20.23540   8.989 

0.15 

16.57120  13.476 

17.41945  13.479 

18.31130  13.481 

19.24890  13.483 

20.23460  13.484 

O.2 

16.56996  17.969 

17.41830  17-972 

18.31020  17.975 

19.24790  17.977 

20.23365  17.979 

O.25 

16.56840  22.463 

17.41680  22.467 

18.30880  22.470 

19.24650  22.473 

20.23240  22.475 

O-3 

16.56660  26.958 

17.41510  26.962 

18.30715  26.965 

19.24496   26.969 

20.23090  26.972 

0-35 

16.56460  31.454 

17.41310  31.458 

18.30530  31.462 

19.24323  31.466 

20.22930  31.469 

0.4 

16.56240  35.950 

17.41110  35.955 

18-30335  35-959 

19.24134  35.963 

20.22750  35.967 

0.45 

16.56010  40.448 

17.40900  40.453 

18.30126  40.458 

19.23940  40.462 

20.22560  40.465 

o-5 

16.55770  44-948 

17.40665  44-953 

18.29910  44-957 

19.23732  44.961 

20.22365  44-965 

0-55 

16.55538  49-448 

17.40441  49-453 

18.29699   49.458 

19.23530  49.462 

20.22170  49.465 

0.6 

16.55307  53-950 

17.40222   53.955 

18.29490  53-959 

19-23331  53-963 

20.21981  53.967 

0.65 

16.55087  58.454 

17.40015   58-458 

18.29292   58.462 

19.23142  58.465 

20.21804  58.469 

0.7 

16.54885  62.958 

17.39822  62.962 

18.29109  62.965 

19.22968  62.969 

20.21639  62.972 

0-75 

16.54702  67.463 

17.39650  67.466 

18.28942  67.470 

19.22813  67.473 

20.21490  67.475 

0.8 

16.54550  71-969 

17.39504  71-972 

18.28805   71-975 

19.22681   71.977 

20.21365  71-979 

0.85 

16.54428  76.476 

17.39386   76.479 

18.28694  76.481 

19.22577  76.483 

20.21268  76.484 

0.9 

16.54337  80.984 

17.39300  80.985 

18.28611   80.987 

19.22496   80.988 

20.21189  80.989 

0-95 

16.54281  85.492 

17.39248  85.493 

18.28560  85.493 

19.22448  85.494 

20.21140  85.495 

I.O 

16.54263  90.000 

17.39230  90.000 

18.28546  90.000 

19.22434  90.000 

20.21129  90.000 

1-05 

16.54281  94.508 

17.39248  94.507 

18.28560  94.507 

19.22448  94.506 

20.21140  94.506 

i.i 

16.54337  99.016 

17.39300  99.015 

18.28611  99.013 

19.22496  99.012 

20.21189  99.011 

1.15 

16.54428  103.524 

17.39386  103.522 

18.28694  103.519 

19.22577  103.518 

20.21268  103.516 

1.2 

16.54550  108.031 

I7-395O4  108.028 

18.28805  108.025 

19.22681  108.023 

20.21365  108.021 

•25 

16.54702  112.537 

17.39650  112.534 

18.28942  112.530 

19.22813  112.527 

20.21490  112.525 

•3 

16.54885  117.042 

17.39822  117.038 

18.29109  117-035 

19.22968  117.031 

20.21639  117.028 

•35 

16.55087  121.547 

17.40015  121.542 

18.29292  121.538 

19.23142  121.535 

20.21804  121.531 

•4 

16.55307  126.050 

17.40222  126.045 

18.29490  126.041 

19.23331  126.037 

20.21981  126.033 

•45 

16.55538  130.552 

17.40441  130.547 

18.29699  130.542 

19.23530  130.538 

20.22170  130.535 

•5 

16.55770  135.052 

17.40665  135.047 

18.29910  135.043 

19.23732  135.039 

20.22365  135.035 

•55 

16.56010  139.552 

17.40900  139.547 

18.30126  139.542 

19.23940  139.538 

20.22560  139.535 

.6 

16.56240  144.050 

17.41110  144-045 

18.30335  144.041 

19.24134  144.037 

20.22750  144.033 

•65 

16.56460  148.546 

17.41310  148.542 

18.30530  148.538 

19.24323  148.534 

20.22930  148.531 

•7 

16.56660  153.042 

17.41510  153.038 

18.30715  153.035 

19.24496  153.031 

20.23090  153.028 

•75 

16.56840  157-537' 

17.41680  157-533 

18.30880  157.530 

19.24650  157-527 

20.23240  157.525 

.8 

16.56996  162.031 

17.41830  162.028 

18.31020  162.025 

19.24790  162.023 

20.23365  162.021 

1.85 

16.57120  166.524 

17.41945  166.521 

18.31130  166.519 

19.24890  166.517 

20.23460  166.516 

1.9 

16.57210  171.016 

17.42030  171.015 

18.31210  171.013 

19.24970  171.012 

20.23540  171.011 

i-95 

16.57260  175.508 

17.42083  175-507 

18.31260  175-507 

19.25015  175.506 

20.23585  175-505 

2.O 

16.57282  180.000 

17.42102  180.000 

18.31278  180.000 

19-25033  180.000 

20.23601  180.000 

Example.      cosh  (3.65  +  i  0.05)  =  19.25015  /4°.494  =  19.25015  /4°.29/.38*. 


[120] 


TABLE  XL    HYPERBOLIC  COSINES,     cosh  (x  +  iq)  =  r  /r    CONTINUED 


x  =  3-75 


*  =  3- 


=  3-8S 


=  3-95 


0 

21.27230   o.ooo 

22.36178   o.ooo 

23-50717   o.ooo 

24.71135   o.ooo 

25-9773I   o.ooo 

0.05 

21.27212     4.495 

22.36163   4.495 

23.50702   4.496 

24.71120   4.496 

25-97720   4.497 

O.I 

2I.27I7O    8.990 

22.36122   8.991 

23.50664   8.992 

24.71090   8.993 

25.97680   8.993 

0.15 

2I.27IOO    13.486 

22.36055  13.487 

23.50600  13.488 

24.71024  13.489 

25-97630  13.490 

0.2 

2I.270OO   17.981 

22.35962  17-983 

23.50512  17.985 

24.70940  17.986 

25-97550  I7-988 

0.2S 

21.26885    22.478 

22.35850  22.480 

23.50404  22.482 

24.70840  22.483 

25-97450  22.485 

o-3 

21.26745    26.974 

22.35716  26.977 

23.50277  26.979 

24.70720  26.981 

.  25.97337  26.983 

o-3S 

21.26586   31.472 

22.35565  31-474 

23-50136  31-477 

24.70580  31.479 

25.97206  31.481 

0.4 

21.26415   35.970 

22.35403  35-973 

23.49980  35.975 

24.70440  35.978 

25.97066  35.980 

o-4S 

21.26236   40.469 

22.35230  40.472 

23.49820  40.474 

24.70280  40.477 

25.96920  40.479 

o-5 

2I.26O52   44.968 

22.35060  44-971 

23.49650  44-974 

24.70120  44.977 

25.96770  44-979 

o-55 

21.25869   49.469 

22.34883  49-472 

23.49486  49.474 

24.69964  49.477 

25.96618  49.479 

0.6 

21.25685    53.970 

22.34712   53.973 

23.49322   53.975 

24.69809  53.978 

25.96471   53.980 

0.65 

21.25520   58.472 

22.34550  58.474 

23-49II5   58.477 

24.69662  58.479 

25-96333   58-481 

0.7 

21.25362   62.974 

22.34401  62.977 

23.49028  62.979 

24.69528  62.981 

25.96205   62.983 

0-75 

21.25221    67.478 

22.34270  67.480 

23.48900   67.482 

24.69406  67.483 

25.96090  67.485 

0.8 

21.25102    71.981 

22.34153   71-983 

23.48791   71.985 

24.69302  71.986 

25.95991   71.988 

0.85 

2I.250O6    76.486 

22.34061   76.487 

23.48704   76.488 

24.69223  76.489 

25.95911   76.490 

0.9 

21.24935   80.990 

22.33995  80.991 

23.48640  80.992 

24.69159  80.993 

25-95854   80.993 

o-95 

21.24891    85.495 

22.33952  85.496 

23.48601   85.496 

24.69120  85.496 

25.95820  85.497 

I.O 

21.24878   90.000 

22.33941   90.000 

23.48589   90.000 

24.69110  90.000 

25.95806   90.000 

1.05 

21.24891    94-50S 

22.33952  94-504 

23.48601   94.504 

24.69120  94.504 

25.95820  94-503 

i.i 

21.24935   99-010 

22-33995  99-009 

23.48640  99.008 

24.69159  99.007 

25-95854  99-007 

i-iS 

21.25006  103.514 

22.34061  103.513 

23.48704  103.512 

24.69223  103.511 

25.95911  103.510 

1.2 

2I.25IO2  IO8.OI9 

22.34153  108.017 

23.48791  108.015 

24.69302  108.014 

25-9599I  108.013 

I-2S 

21.25221   112.522 

22.34270  112.520 

23.48900  112.518 

24.69406  112.517 

25.96000  112.515 

i-3 

21.25362  II7.O26 

22.34401  117.023 

23.49028  117.021 

24.69528  117.019 

25.96205  117.017 

i-3S 

2I.2552O  121.528 

22.34550  121.526 

23.49115  121.523 

24.69662  121.521 

25-96333  121.519 

1.4 

21.25685  I26.O3O 

22.34712  126.027 

23.49322  126.025 

24.69809  126.022 

25.96471  126.020 

i.  45 

21.25869  130.531 

22.34883  130.528 

23.49486  130.525 

24.69964  130.523 

25.96618  130.521 

i-S 

21.26052  135.032 

22.35060  135.029 

23.49650  135.026 

24.70120  135.023 

25.96770  135.021 

i-55 

21.26236  I39-531 

22.35230  139.528 

23.49820  139.526 

24.70280  139.523 

25.96920  139.521 

1.6 

21.26415  144.030 

22.35403  144.027 

23.49980  144-025 

24.70440  144.022 

25.97066  144.020 

1.65 

21.26586  148.528 

22.35565  148.526 

23.50136  148.523 

24.70580  148.521 

25.97206  148.519 

i-7 

21.26745  153.026 

22.35716  153.023 

23-50277  153-021 

24.70720  153.019 

25-97337  I53-OI7 

i-75 

21.26885  I57-522 

22.35850  157.520 

23.50404  157-518 

24.70840  157.517 

25-9745°  I57-5I5 

1.8 

21.27000  I62.OI9 

22.35962  162.017 

23.50512  162.015 

24.70940  162.014 

25.97550  162.012 

1.85 

2I.27IOO  166.514 

22.36055  166.513 

23.50600  166.512 

24.71024  166.511 

25.97630  166.510 

1.9 

2I.27I7O  I7I.OIO 

22.36122  171.009 

23.50664  171.008 

24.71090  171.007 

25.97680  171.007 

i-9S 

21.27212  I75-505 

22.36163  175-505 

23.50702  175-504 

24.71120  i75-5°4 

25.97720  175.503 

2.0 

2I.2723O  iSo.OOO 

22.36178  180.000 

23.50717  180.000 

24.71135  180.000 

25-9773I  180.000 

Example.      cosh  (3.85  +  i  1.05)  =  23.48601 


=  23.48601  /94°.3o/.i4>. 


[121] 


TABLE  XII.     HYPERBOLIC  TANGENTS,     tanh  (x  +  iq)  =  r 


*  = 

0 

X  = 

0.05 

*  = 

O.I 

*  = 

3.15 

X  = 

O.2 

9 

r 

7 

r 

7 

f 

7 

r 

7 

r 

7 

0 

o 

o 

0 

0 

0 

0.00 

90 

0.04996 

o.oo 

0.09967 

O.OO 

0.14889 

O.OO 

0.19738 

O.OO 

0.05 

0.07870 

9P 

0.09322 

57.368 

0.12699 

37.846 

0.16840 

27.190 

0.21246 

20.849 

O.I 

0.15838 

90 

0.16607 

72.040 

O.I87II 

56.914 

0.21732 

45.420 

0.25294 

36.955 

0.15 

0.24008 

90 

0.24520 

77.558 

0.25987 

66.084 

0.28232 

56.148 

0.31045 

47.863 

0.2 

0.32492 

90 

0.32870 

80.329 

0.33969 

71.092 

0-35699 

62.612 

0-37939 

55.054 

0.25 

0.41421 

90 

0.41727 

81.937 

0.42568 

74.107 

0-43932 

66.700 

0.45731 

59.848 

0-3 

0.50953 

9° 

0.51180 

82.942 

0.51851 

76.025 

0.52931 

69.373 

0.54368 

63.082 

o-3S 

0.61280 

90 

0.61455 

83.585 

0.61970 

77.267 

0.62802 

71.131 

0.63915 

65.250 

0.4 

0.72654 

90 

0.72778 

83.987 

o.73J43 

78.047 

0-73734 

72.245 

0.74525 

66.641 

0-45 

0.85408 

90 

0.85476 

84.209 

0.85678 

78.478 

0.86004 

72.864 

0.86439 

67.419 

0-5 

I.OOOOO 

90 

I.OOOOO 

84.279 

I.OOOOO 

78.616 

I.OOOOO 

73.064 

I.OOOOO 

67.670 

0-5S 

1.17085 

90 

1.16991 

84.209 

1.16717 

78.478 

1.16274 

72.864 

1.15688 

67.419 

0.6 

1.37638 

90 

1.37404 

83.987 

1.36718 

78.047 

1.35623 

72.245 

1-34183 

66.641 

0.65 

1.63185 

90 

1.62722 

83.585 

1.61369 

77.267 

1.59231 

71.131 

1.56459 

65.250 

0.7 

1.96261 

90 

1.95388 

82.942 

1.92859 

76.025 

1.88925 

69.373 

I.83933 

63.082 

0-75 

2.41421 

90 

2.39735 

81.937 

2.34919 

74.107 

2.27623 

66.700 

2.18669 

59.848 

0.8 

3.07768 

90 

3-04234 

80.329 

2.94391 

71.092 

2.80120 

62.612 

2.63581 

55-054 

0.85 

4.16530 

90 

4.07824 

77.558 

3.84810 

66.084 

3.54212 

56.148 

3-22115 

47-863 

0.9 

6.31375 

90 

6.02149 

72.040 

5.34442 

56.914 

4.60155 

45.420 

3-95347 

36.955 

o-9S 

12.70620 

90 

10.72750 

57.368 

7.87464 

37.846 

5-93842 

27.190 

4.70673 

20.849 

I.O 

00 

90 

20.01667 

G.OO 

10.03331 

O.OO 

6.71659 

O.OO 

5.06649 

O.OO 

1.05 

12.70620 

90 

10.72750 

57-368 

7.87464 

37.846 

5-93842 

27.190 

4.70673 

20.849 

.1 

6.31375 

90 

6.02149 

72.O4O 

5.34442 

36.914 

4.60155 

45.420 

3-95347 

36.955 

•IS 

4.16530 

90 

4.07824 

77.558 

3.84810 

66.084 

3-54212 

56.148 

3-22115 

47.863 

.2 

3-07768 

90 

3-04234 

80.329 

2.94391 

71.092 

2.80120 

62.612 

2.63581 

55.054 

•25 

2.41421 

90 

2-39735 

81.937 

2.34919 

74.107 

2.27623 

66.700 

2.18669 

59.848 

•3 

1.96261 

90 

1.95388 

82.942 

1.92859 

76.025 

1.88925 

69.373 

I-83933 

63.082 

•35 

1.63185 

90 

1.62722 

83.585 

1.61369 

77.267 

1-59231 

71.131 

1.56459 

65.250 

1.4 

1-37638 

90 

1.37404 

83.987 

1.36718 

78.047 

1.35623 

72.245 

1-34183 

66.641 

1-45 

1.17085 

90 

1.16991 

84.209 

1.16717 

78.478 

1.16274 

72.864 

1.15688 

67.419 

•5 

I.OOOOO 

90 

I.OOOOO 

84.279 

I.OOOOO 

78.616 

I.OOOOO 

73.064 

1.09000 

67.670 

•55 

0.85408 

90 

0.85476 

84.209 

0.85678 

78.478 

0.86004 

72.864 

0.86439 

67.419 

.6 

0.72654 

90 

0.72778 

83-987 

0.73*43 

78.047 

0.73734 

72.245 

0.74525 

66.641 

•65 

0.61280 

90 

0.61455 

83.585 

0.61970 

77.267 

0.62802 

71.131 

0.63915 

65.250 

•7 

0.50953 

90 

0.51180 

82.942 

0.51851 

76.023 

0.52931 

69.373 

0.54368 

63.082 

•75 

0.41421 

90 

0.41727 

81.937 

0.42568 

74.107 

0.43932 

66.700 

0.45731 

59.848 

.8 

0.32492 

90 

0.32870 

80.329 

0.33969 

71.092 

0.35699 

62.612 

0-37939 

55.054 

•85 

0.24008 

90 

0.24520 

77.558 

0.25987 

66.084 

0.28232 

56.148 

0.31045 

47.863 

•9 

0.15838 

9P 

0.16607 

72.O4O 

0.18711 

56.914 

0.21732 

45.420 

0.25294 

36.953 

•95 

0.07870 

90 

0.09322 

57.368 

0.12699 

37.846 

0.16840 

27.190 

0.21246 

20.849 

2.0       o.oo        90      0.04996     o.oo        0.09967     o.oo        0.14889     o.oo        0.19738     o.oo 

Note.  Negative  quantities  are  in  heavy  type. 

Examples,     tanh  (o.i  +  i  0.25)  =  0.42568  /74°.  107  =  0.42568  /74°.o6'.2$".m 
tanh  (o.i  +  i  1.2)  =  2.94391  \7i°.o92  =  2.94391  \7i°.o5'.3i*. 


[122] 


TABLE  XII.     HYPERBOLIC  TANGENTS,     tanh  (x  +  iq)  =  r 


*  =  0.25 


x  =  0.3 


=  o-3S 


x  =  0.4 


CONTINUED 


=  0.45 


q 

r 

V 

r 

7 

r 

7 

f 

7 

r 

7 

o 

0 

o 

o 

o 

o 

0.24492 

o.oo 

0.29131 

O.OO 

0.33638 

o.oo 

0.37995 

o.oo 

0.42190 

0.00 

0.05 

0.25721 

16.710 

0.30168 

13-805 

0-34534 

11.652 

0.38784 

9.990 

0.42894 

8.665 

O.I 

0.29145 

30.669 

0.33123 

25.891 

0.37127 

22.164 

0.41000 

19.185 

0.44965 

16.753 

0.15 

0.34237 

41.063 

0.37657 

35-492 

0.41192 

30.900 

0.44758 

27.076 

0.48295 

23.858 

O.2 

0.40561 

48.442 

0-43445 

42.715 

0.46491 

37-770 

0.49617 

33-498 

0.52758 

29.796 

0.2S 

0.47875 

53-612 

0.50275 

48.001 

0.52849 

42.989 

0.55525 

38.527 

0.58242 

34-560 

0-3 

0.56098 

57-214 

0.58056 

51-799 

0.60177 

46.843 

0.62401 

42.331 

0.64675 

38.242 

0-35 

0.65262 

59-679 

0.66796 

54-453 

0.68466 

49-590 

0.70225 

45-093 

0.72031 

40.958 

0.4 

0.75486 

61.281 

0.76580 

56.200 

0.77774 

51-423 

0.79033 

46.961 

0.80327 

42.815 

0-45 

0.86968 

62.184 

0.87570 

57-194 

0.88225 

52.474 

0.88914 

48.039 

0.89620 

43.896 

0-5 

I.OOOOO 

62.476 

I.OOOOO 

57.518 

I.OOOOO 

52.817 

I.OOOOO 

48.392 

I.OOOOO 

44.250 

o-55 

1.14985 

62.184 

1.14195 

57-194 

1-13347 

52.474 

1.12469 

48.039 

1.11583 

43.896 

0.6 

1.32476 

61.281 

1.30582 

56.200 

1.28577 

51-423 

1.26529 

46.961 

1.24492 

42.815 

0.65 

1.53228 

59-679 

1.49710 

54-453 

1.46059 

49.590 

1.42400 

45.093 

1.38830 

40.958 

0.7 

1.78259 

57-214 

1.72246 

51-799 

1.66176 

46.843 

1.60255 

42.331 

1.54620 

38.242 

0-75 

2.08878 

53-612 

1.98907 

48.001 

1.89219 

42.989 

1.80100 

38.527 

1.71698 

34-560 

0.8 

2.46545 

48.442 

2.30177 

42.715 

2.15096 

37-770 

2.01545 

33-498 

1.89545 

29.796 

0.85 

2.92081 

41.063 

2-65553 

35-492 

2.42764 

30.900 

2.23422 

27.076 

2.07060 

23.858 

0.9 

3-43113 

30.669 

3.01903 

25.891 

2.69344 

22.164 

2.43370 

19.185 

2.22397 

16.753 

0-95 

3.88795 

16.710 

3.31480 

13-805 

2.89571 

11.652 

2.57838 

9.990 

2.33132 

8.665 

I.O 

4.08299 

0.00 

3-43274 

o.oo 

2.97287 

o.oo 

2.63193 

o.oo 

2.37024 

o.oo 

1.05 

3.88795 

16.710 

3.31480 

13.805 

2.89571 

11.652 

2.57838 

9.990 

2.33132 

8.665 

i.i 

3-43113 

30.669 

3.01903 

25.891 

2.69344 

22.164 

2.43370 

19-185 

2.22397 

16.753 

i-iS 

2.92081 

41.063 

2.65553 

35.492 

2.42764 

30.900 

2.23422 

27.076 

2.07060 

23.858 

1.2 

2.46545 

48.442 

2.30177 

42.715 

2.15096 

37.770 

2.01545 

33-498 

1.89545 

29.796 

1.25 

2.08878 

53.612 

1.98907 

48.001 

1.89219 

42.989 

1.80100 

38.527 

1.71698 

34.560 

i-3 

1.78259 

57.214 

1.72246 

51-799 

1.66176 

46.843 

1.60255 

42.331 

1.54620 

38.242 

i-3S 

1.53228 

59.679 

1.49710 

54-453 

1.46059 

49.590 

1.42400 

45.093 

1.38830 

40.958 

1.4 

1.32476 

61.281 

1.30582 

56.200 

1.28577 

51-423 

1.26529 

46.961 

1.24492 

42.815 

i-4S 

1.14985 

62.184 

1.14195 

57-194 

1-13347 

52.474 

1.12469 

48.039 

1.11583 

43.896 

i-5 

I.OOOOO 

62.476 

I.OOOOO 

57.518 

I.OOOOO 

52.817 

I.OOOOO 

48.392 

I.OOOOO 

44.250 

i-55 

0.86968 

62.184 

0.87570 

57-194 

0.88225 

52.474 

0.88914 

48.039 

0.89620 

43.896 

1.6 

0.75486 

61.281 

0.76580 

56.200 

0.77774 

51-423 

0.79033 

46.961 

0.80327 

42.815 

1.65 

0.65262 

59-679 

0.66796 

54-453 

0.68466 

49.590 

0.70225 

45.093 

0.72031 

40.958 

i-7 

0.56098 

57-214 

0.58056 

51-799 

0.60177 

46.843 

0.62401 

42.331 

0.64675 

38.242 

i-7S 

0.47875 

53-612 

0.50275 

48.001 

0.52849 

42.989 

0.55525 

38.527 

0.58242 

34.560 

1.8 

0.40561 

48.442 

0.43445 

42.715 

0.46491 

37.770 

0.49617 

33.498 

0.52758 

29.796 

1.85 

0.34237 

41.063 

0.37657 

35-492 

0.41192 

30.900 

0.44758 

27.076 

0.48295 

23.858 

1.9 

0.29145 

30.669 

0.33123 

25.891 

0.37127 

22.164 

0.41090 

19.185 

0.44965 

16.753 

i.  95 

0.25721 

16.710 

0.30168 

13-805 

0.34534 

11.652 

0.38784 

9.990 

0.42894 

8.665 

2.0        0.24492     o.oo        0.29131     o.oo        0.33638     o.oo        0.37995     o.oo        0.42190     o.oo 

Note.  Negative  quantities  are  in  heavy  type. 

Examples,     tanh  (0.4  +  i  o)  =  0.37995  /o° 


tanh  (0.45  +  *  i.i)  =  2.22397  \i6°.753  =  2.22397  \i6°.45'.ii*. 


[123] 


TABLE  XII.     HYPERBOLIC  TANGENTS,     tanh  (*  +  *?)  =  r /y.     CONTINUED 


*  =  o.s 

*  =  0.55 

*  =  0.6 

x  =  0.65 

*  =  0.7 

q 

r      y 

r     7 

r     7 

r     7 

r     7 

0 

o 

o 

o 

0 

o 

0.46211  o.oo 

0.50052  o.oo 

0.53704  o.oo 

0.57167  o.oo 

0.60437  o.oo 

0.05 

0.46846  7.582 

0.50628  6.680 

0.54230  5.917 

0.57648   5.263 

0.60878  4.696 

O.I 

0.48720  14-732 

0.52334  13.027 

0.55790  11.570 

0.59079  10.312 

0.62194  9.217 

0.15 

0.51758  21.122 

0.55115  18.773 

0.58344  16.739 

0.61428  14.966 

0.64357  13.409 

0.2 

0.55865  26.572 

0.58900  23.753 

0.61835  21.276 

0.64650  19.090 

0.67331  17.153 

0.2S 

0.60952  31.035 

0.63616  27.897 

0.66205  25.101 

"0.68696  22.604 

0.71076  20.372 

0-3 

0.66956  34-544 

0.69209  31.204 

0.71405  28.190 

0.73523  25.471 

0.75548  23.017 

0-35 

0.73847  37.I69 

0.75645  33.707 

0.77399  30.553 

0.79092  27.683 

0.80711  25.074 

0.4 

0.8l62*  38.983 

0.82914  35.453 

0.84168  32.214 

0.85377  29.248 

0.86531  26.539 

0-45 

0.90328  40.046 

0.91025  36.482 

0.91703  33.J98 

0.92354  3°.l8o 

0.92973  27.414 

0-5 

1.  00000  40.395 

i.ooooo  36.822 

i.ooooo  33.524 

i.ooooo  30.489 

i.ooooo  27.705 

0-55 

1.10708  40.046 

1.09860  36.482 

1.09048  33.198 

1.08279  30.180 

1.07558  27.414 

0.6 

1.22508  38.983 

1.20607  35-453 

1.  18810  32.214 

1.17128  29.248 

1.15566  26.539 

0.65 

1-35414  37.169 

1-32197  33.707 

1.29201  30.553 

1.26434  27.683 

1.23898  25.074 

0.7 

1-49352  34-544 

1.44490  31.204 

1.40047  28.190 

1.36012  25.471 

1.32366  23.017 

0-75 

1.64064  31.035 

1.57193  27.897 

1.51047  25.101 

1.45568  22.604 

1.40695  20.372 

0.8 

1.79004  26.572 

1.69780  23.753 

1.61722  21.276 

1.54680  19.090 

1.48519  17.153 

0.85 

1.93206  21.122 

1.81438  18.773 

1.71398  16.739 

1.62793  14.966 

1.55384  13.409 

0.9 

2.05254  14.732 

1.91081  13.027 

1.79243  11.570 

1.69266  10.312 

1.60788  9.217 

o-9S 

2.13465    7.582 

1.97520  6.680 

1.84400  5.917 

1.73467  5.263 

1.64262  4.696 

1.0 

2.16395  o.oo 

1.99792  o.oo 

1.86202  o.oo 

1.74926  o.oo 

1.65462  o.oo 

1.05 

2.13465  7.582 

1.97520  6.680 

1.84400  5.917 

1.73467  5.263 

1.64262  4.696 

i.i 

2.05254  14-732 

1.91081  13.027 

1.79243  11.570 

1.69266  10.312 

1.60788  9.217 

i.iS 

1.93206  21.122 

1.81438  18.773 

1.71398  16.739 

1.62793  14.966 

1.55384  13.409 

1.2 

1.79004  26.572 

1.69780  23.753 

1.61722  21.276 

1.54680  19.090 

1.48519  17.153 

I-2S 

1.64064  31.035 

1.57193  27-897 

1.51047  25.101 

1.45568  22.604 

1.40695  20.372 

i-3 

1.49352  34-544 

1.44490  31.204 

1.40047  28.190 

1.36012  25.471 

1.32366  23.017 

i-3S 

1.35414  37-169 

1.32197  33-707 

1.29201  30.553 

1.26434  27.683 

1.23898  25.074 

1.4 

1.22508  38.983 

1.20607  35-453 

I.  18810  32.214 

1.17128  29.248 

1.15566  26.539 

1-45 

1.10708  40.046 

1.09860  36.482 

1.09048  33.198 

1.08279  30.180 

1.07558  27.414 

i-S 

i.ooooo  40.395 

i.ooooo  36.822 

i.ooooo  33.524 

i.ooooo  30.489 

i.ooooo  27.705 

i-SS 

0.90328  40.046 

0.91025  36.482 

0.91703  33-198 

0.92354  30.180 

0.92973  27.414 

1.6 

0.81628  38.983 

0.82914  35.453 

0.84168  32.214 

0.85377  29.248 

0.86531  26.539 

1.65 

0.73847  37-i69 

0.75645  33.707 

0.77399  30.553 

0.79092  27.683 

0.80711  25.074 

i-7 

0.66956  34-544 

0.69209  31.204 

0.71405  28.190 

0.73523  25.471 

0.75548  23.017 

i-7S 

0.60952  31-035 

0.63616  27.897 

0.66205  25.101 

0.68696  22.604 

0.71076  20.372 

1.8 

0.55865  26.572 

0.58900  23.753 

0.61835  21.276 

0.64650  19.090 

0.67331  17-153 

1.85 

0.51758  21.122 

0.55115  18.773 

0.58344  16.739 

0.61428  14.966 

0.64357  13.409 

1.9 

0.48720  14.732 

0.52334  13.027 

0.55790  11.570 

0.59079  10.312 

0.62194  9.217 

1-95 

0.46846   7.582 

0.50628   6.680 

0.54230  5.917 

0.57648  5.263 

0.60878  4.696 

2.0        0.46211     o.oo        0.50052     o.oo        0.53704     o.oo        0.57167     o.oo        0.60437     o.oo 

Note.  Negative  quantities  are  in  heavy  type. 

Examples,     tanh  (0.7  +  i  0.7)  =  1.32366  /23°.oi7  =  1.32366  /23°.oi>.oi*'. 
tanh  (0.6  +  »  1.5)  =  i.oooo  \33°.524  =  i.oooo  \33°.3i'.26"'. 


[124] 


TABLE  XII.     HYPERBOLIC  TANGENTS,     tanh  (x  +  ig)  =  r  Ay.     CONTINUED 


x  =  0.75 


x  =  0.85 


x  =  o.g 


x  =  0.95 


0 

0.63515 

o.oo 

0.66403 

0.00 

0.69107 

o.oo 

0.71629 

o.oo 

0.73978 

o.oo 

0.05 

0.63921 

4.202 

0.66777 

3.767 

0.69451 

3.384 

0.71947 

3-043 

0.74269 

2.741 

O.I 

0.65131 

8.257 

0.67892 

7.411 

0.70478 

6.662 

0.72892 

5-995 

0.75141 

5.401 

0.15 

0.67125 

12.036 

0.69730 

10.819 

0.72172 

9-737 

0-74453 

8.771 

0.76578 

7-909 

O.2 

0.69871 

15-432 

0.72264 

13.898 

0.74509 

12.526 

0.76607 

11.297 

0.78561 

10.196 

O.2S 

0-73333 

18.371 

0.75461 

16.576 

0-77459 

14.964 

0.79326 

I3-5I3 

0.81065 

12.208 

0-3 

0.77471 

20.804 

0.79285 

18.807 

0.80986 

17.004 

0.82576 

15-375 

0.84054 

13-904 

0.35 

0.82247 

22.707 

0.83695 

20.559 

0.85051 

18.613 

0.86316 

16.848 

0.87492 

15-250 

0.4 

0.87623 

24.068 

0.88650 

21.819 

0.89611 

19-773 

0.90504 

17.914 

0.91333 

16.226 

0.45 

0-93557 

24.885 

0.94104 

22.576 

0.94614 

20.472 

0.95086 

18.557 

0.95523 

16.816 

0.5 

I.OOOOO 

25-157 

.00000 

22.828 

I.OOOOO 

20.706 

I.OOOOO 

18.772 

I.OOOOO 

17.013 

0.55 

1.06887 

24.885 

.06265 

22.576 

1.05693 

20.472 

1.05168 

18.557 

1.04687 

16.816 

0.6 

1.14125 

24.068 

.12803 

21.819 

1.11594 

19-773 

1.10492 

17.914 

1.09490 

16.226 

0.65 

1.21585 

22.707 

.19482 

20.559 

1.17576 

18.613 

1.15853 

16.848 

1.14297 

15-250 

0.7 

1.29081 

20.804 

.26128 

18.807 

1.23478 

17.004 

I.2IIOI 

15-375 

1.18971 

13.904 

0-75 

1.36365 

18.371 

1-32519 

16.576 

1.29101 

14.964 

I.26O62 

I3-5I3 

1.23358 

12.208 

0.8 

1.43121 

15-432 

1.38382 

13.898 

1.34212 

12.526 

1.30536 

11.297 

1.27289 

10.196 

0.85 

1.48976 

12.036 

1.43411 

10.819 

L38559 

9-737 

I-343I3 

8.771 

1.30586 

7.909 

0.9 

1-53537 

8.257 

1.47293 

7.411 

1.41889 

6.662 

I.37I89 

5-995 

1.33083 

5.401 

o-9S 

1.56444 

4.202 

I-4975I 

3-767 

1.43986 

3-384 

1.38992 

3-043 

1.34645 

2.741 

I.O 

1-57443 

o.oo 

I.50594 

o.oo 

1.44703 

o.oo 

1.39606 

o.oo 

1.35175 

o.oo 

1.05 

1.56444 

4.202 

J-49751 

3.767 

1.43986 

3.384 

1.38992 

3.043 

1.34645 

2.741 

i.i 

1-53537 

8.257 

1.47293 

7.411 

1.41889 

6.662 

I.37I89 

5-995 

1.33083 

5.401 

i-i5 

1.48976 

12.036 

1.43411 

10.819 

I.38559 

9-737 

L343I3 

8.771 

1.30586 

7.909 

1.2 

1.43121 

15.432 

1.38382 

13.898 

1.34212 

12.526 

1.30536 

11.297 

1.27289 

10.196 

1-25 

1-36365 

18.371 

1-325*9 

16.576 

.29101 

14.964 

1.26062 

I3.5U 

1-23358 

12.208 

i-3 

1.29081 

20.804 

1.26128 

18.807 

•23478 

17.004 

I.2IIOI 

15-375 

1.18971 

13.904 

1-35 

1-21585 

22.707 

1.19482 

20.559 

•17576 

18.613 

I.I5853 

16.848 

1.14297 

15.250 

1.4 

1.14125 

24.068 

1.12803 

21.819 

-II594 

19-773 

I.I0492 

17.914 

1.09490 

16.226 

i  -45 

1.06887 

24.885 

1.06265 

22.576 

•05693 

20.472 

I.05I68 

18.557 

1.04687 

16.816 

i-5 

I.OOOOO 

25.157 

I.OOOOO 

22.828 

I.OOOOO 

20.706 

I.OOOOO 

18.772 

I.OOOOO 

17.013 

1-55 

0-93557 

24.885 

0.94104 

22.576 

0.94614 

20.472 

0.95086 

iS-557 

0.95523 

16.816 

1.6 

0.87623 

24.068 

0.88650 

21.819 

0.89611 

19-773 

0.90504 

17.914 

0.91333 

16.226 

1.65 

0.82247 

22.707 

0.83695 

20.559 

0.85051 

18.613 

0.86316 

16.848 

0.87492 

15-250 

i.7 

0.77471 

20.804 

0.79285 

18.807 

0.80986 

17.004 

0.82576 

15-375 

0.84054 

13.904 

i-75 

0-73333 

18.371 

0.75461 

16.576 

0-77459 

14.964 

0.79326 

13-513 

0.81065 

12.208 

1.8 

0.69871 

15.432 

0.72264 

13.898 

0-74509 

12.526 

0.76607 

11.297 

0.78561 

10.196 

1.85 

0.67125 

12.036 

0.69730 

10.819 

0.72172 

9-737 

0.74453 

8.771 

0.76578 

7.909 

1.9 

0.65131 

8.257 

0.67892 

7.411 

0.70478 

6.662 

0.72892 

5-995 

0.75141 

5.401 

i-95 

0.63921 

4.202 

0.66777 

3.767 

0.69451 

3-384 

0.71947 

3.043 

0.74269 

2.741 

2.0   0.63515  o.oo    0.66403  o.oo    0.69107  o.oo    0.71629  o.oo    0.73978  o.oo 


Note. 


Negative  quantities  are  in  heavy  type. 


Examples,     tanh  (0.9  +  i  i.o)  =  1.39606  /o°. 


tanh  (0.95  +  i  1.55)  =  0.95523  \i6°.8i6  =  0.95523  \i6°.48'.s8'. 


TABLE  XII.     HYPERBOLIC  TANGENTS,     tanh  (*  +  iq)  = 


x  =  1.05 


*  =  1.15 


1 

0.76159  o.oo 

0.78181  o.oo 

0.80050  o.oo 

0.81775  °-oo 

0.83365  o.oo 

5.05 

0.76428  2.470 

0.78427  2.228 

0.80277    2.OIO 

0.81984  1.815 

0.83557  1-639 

3.1 

0.77229  4.870 

0.79164  4.394 

0.80954   3.966 

0.82605  3-582 

0.84127  3.235 

'•IS 

0.78552  7.134 

0.80380  6.440 

0.82071    5.815 

0.83630  5.254 

0.85067  4.747 

X2 

0.80376  9.206 

0.82058  8.315 

0.83611    7.513 

0.85043  6.790 

0.86361  6.138 

5.2S 

0.82678  11.032 

0.84172  9.971 

0.85551   9.015 

0.86822  8.151 

0.87991  7.371 

5-3 

0.85425  12.575 

0.86693  11.373 

0.87862  10.288 

0.88939  9.306 

0.89927  8.419 

5-35 

0.88580  13.802 

0.89585  12.492 

0.90509  11.305 

0.91359  10.231 

0.92138  9.258 

3.4 

0.92098  14.694 

0.92802  13.305 

0.93449  12.045 

0.94042  10.904 

0.94585  9.870 

3-45 

0.95925  15.233 

0.96294  13.798 

0.96632  12.495 

0.96941  11.313 

0.97223  10.242 

3-5 

i.ooooo  15.414 

i.ooooo  13.963 

i.ooooo  12.646 

i.ooooo  11.451 

i.ooooo  10.368 

^•55 

1.04248  15.233 

1.03849  13.798 

1.03486  12.495 

1.03155  11.313 

1.02854  10.242 

3.6 

1.08581  14.694 

1.07756  13.305 

1.07010  12.045 

1.06335  10.904 

1.05725  9.870 

x6S 

1.12892  13.802 

1.11626  12.492 

1.10486  11.305 

1.09458  10.231 

1.08533  9-258 

3-7 

1.17061  12.575 

1.15349  11.373 

1.13815  10.288 

1.12437  9.306 

I.II20I    8.419 

3-75 

1.20951  11.032 

1.18804  9.971 

1.16889  9-OI5 

1.15179  8.151 

1.13649  7.371 

D.8 

1.24415  9.206 

1.21866  8.315 

1.19602   7.513 

1.17587  6.790 

1.15793  6.138 

3.85 

1-27305  7-134 

1.24409  6.440 

1.21846  5.815 

1.19574  5.254 

I-I7555  4-747 

D.9 

1.29485  4.870 

1.26320  4-394 

i-23527  3-966 

1.21058  3.582 

1.18868  3.235 

3-95 

1.30843  2.470 

1.27506   2.228 

1.24569    2.OIO 

1.21976   1.815 

1.19680  1.639 

i.o 

1.31304  o.oo 

1.27008  o.oo 

1.24922  o.oo 

1.22286  o.oo 

1.19954  o.oo 

i-°S 

1.30843  2.470 

1.27506  2.228 

1.24569   2.0IO 

1.21976  1.815 

1.19680  1.639 

i.i 

1.29485  4.870 

1.26320  4.394 

1.23527  3.966 

1.21058  3.582 

1.18868  3.235 

•IS 

1-27305  7.134 

1.24409  6.440 

1.21846  5.815 

1.19574  5.254 

I-I7555  4-747 

.2 

1.24415  9.206 

1.21866  8.315 

1.19602  7.513 

1.17587  6.790 

1.15793  6.138 

•25 

1.20951  11.032 

1.18804  9.971 

1.16889   9'015 

1.15179  8.151 

1.13649  7.371 

•3 

1.17061  12.575 

1.15349  11.373 

1.13815  10.288 

1.12437  9.306 

I.II20I    8.419 

•35 

1.12892  13.802 

1.11626  12.492 

1.10486  11.305 

1.09458  10.231 

I-08533    9.258 

•4 

1.08581  14.694 

1.07756  13.305 

1.07010  12.045 

1.06335  10.904 

I.O5725    9.870 

•45 

1.04248  15.233 

1.03849  13.798 

1.03486  12.495 

1.03155  11.313 

I.O2854  IO.242 

•5 

i.ooooo  15.414 

i.ooooo  13.963 

i.ooooo  12.646 

i.ooooo  11.451 

i.ooooo  10.368 

•55 

0.95925  15.233 

0.96294  13.798 

0.96632  12.495 

0.96941  11.313 

0.97223  10.242 

.6 

0.92098  14.694 

0.92802  13.305 

0.93449  12.045 

0.94042  10.904 

0.94585  9.870 

•65 

0.88580  13.802 

0.89585  12.492 

0.90509  11.305 

0.91359  10.231 

0.92138  9.258 

•7 

0.85425  12.575 

0.86693  "-373 

0.87862  10.288 

0.88939  9.306 

0.89927  8.419 

•75 

0.82678  11.032 

0.84172  9.971 

0.85551  9.015 

0.86822  8.151 

0.87991  7.371 

.8 

0.80376  9.206 

0.82058  8.315 

0.83611  7.513 

0.85043  6.790 

0.86361  6.138 

•85 

0.78552  7.134 

0.80380  6.440 

0.82071  5.815 

0.83630  5.254 

0.85067  4.747 

•0 

0.77229  4.870 

0.79164  4.394 

0.80954  3.966 

0.82605  3.582 

0.84127  3.235 

1-95 

0.76428  2.470 

0.78427   2.228 

O.80277   2.OIO 

0.81984  1.815 

0.83557   »-°39 

2.0        0.76159     o.oo        0.78181     o.oo        0.80050     o.oo        0.81775     o.oo        0.83365     o.oo 

Note.  Negative  quantities  are  in  heavy  type. 

Examples,     tanh  (i.i  +  i  0.7)  =  1.13815  /io°.288  =  1.13815  /io°.i7/.i7*. 


tatth  (1.2  +  f 


0.89927  \8°.4i9  =  0.89927  \8°.2s'.o8*. 


[126] 


TABLE  XII.     HYPERBOLIC  TANGENTS,     tanh  (x  +  iq)  =  r 


CONTINUED 


*  =  1.25 

*  =  1.3 

*  =  i-35 

x  =  1.4 

x  =  1-45 

9 

f     t 

r    y 

f     y 

f     7 

r    7 

o 

o 

o 

o 

0 

0 

0.84828  o.oo 

0.86172  o.oo 

0.87405  o.oo 

0.88535  o.oo 

0.89569  o.oo 

o.os 

0.85004  1.481 

0.86333  1.338 

0.87552  1.  210 

0.88669  I-O94 

0.89692  0.989 

O.I 

0.85526  2.924 

0.86811  2.643 

0.87990  2.389 

0.89069  2.161 

0.90057  1.953 

0.15 

0.86387  4.291 

0-87599  3-879 

O.887II  3.508 

0.89728  3.172 

0.90659  2.869 

O.2 

0.87573  5-549 

0.88684  5-017 

0.89702  4.538 

0.90634  4.104 

0.91485  3.712 

0.25 

0.89063  6.666 

0.90047  6.029 

0.90947  5.454 

0.91769  4.934 

0.92521  4.463 

0-3 

0.90833  7.616 

0.91663  6.890 

O.9242I  6.234 

0.93114  5.640 

0.93746  5.103 

0-35 

0.92852  8.378 

0.93504  7.581 

0.94099  6.86O 

0.94642  6.207 

0-95*37  5-6i7 

0.4 

0.95082  8.933 

0.95534  8.085 

0-95947  7-3I7 

0.96323  6.622 

0.96665  5.993 

0-45 

0.97481  9.272 

0.97715  8.393 

0.97928  7.596 

0.98122  6.875 

0.98298  6.222 

o-S 

i.ooooo  9.385 

i.ooooo  8.496 

i.ooooo  7.689 

i.ooooo  6.960 

i.ooooo  6.299 

o-SS 

1.02584  9.272 

1.02338  8.393 

1.02116  7.596 

1.01914  6.875 

I.OI733  6.222 

0.6 

1.05173  8.933 

1.04674  8.085 

1.04224  7.317 

1.03817  6.622 

1.03450  5.993 

0.65 

1.07699  8.378 

1.06948  7.581 

1.06271  6.860 

1.05661  6.207 

1.05112  5.617 

0.7 

1.10092  7.616 

1.09096  6.890 

1.08200  6.234 

1.07395  5-640 

1.06671  5.103 

0-75 

1.12280  6.666 

1.11054  6.029 

1.09955  5-454 

1.08963  4.934 

1.08084  4-463 

0.8 

1.14192  5.549 

1.12760  5.017 

1.11483  4.538 

1.10335  4.104 

1.09308  3.712 

0.85 

1.15758  4-291 

1.14156  3.879 

1.12726  3.508 

1.11448  3.172 

1.10304  2.869 

0.9 

1.16924  2.924 

1.15193  2.643 

1.13650  2.389 

1.12272  2.161 

1.11041  1.953 

0-95 

1.17642  1.481 

1.15831  1.338 

I.I42I8  1.  210 

1.12779  1-094 

1.11493  0.989 

I.O 

.17885  o.oo 

1.16047  o.oo 

1.14410  o.oo 

1.12950  o.oo 

.11646  o.oo 

1.05 

.17642  1.481 

1.15831  1.338 

I.I42I8  I.2IO 

1.12779  1.094 

.11493  0.989 

.1 

.16924  2.924 

1.15193  2.643 

I.I3650  2.389 

1.12272  2.161 

.11041  1.953 

•IS 

.15758  4.291 

1.14156  3.879 

I.I2726  3.508 

1.11448  3.172 

.10304  2.869 

.2 

.14192  5.549 

1.12760  5.017 

I.II483  4.538 

1.10335  4.104 

.09308  3.712 

•25 

.12280  6.666 

1.11054  6.029 

1.09955  5-454 

1.08963  4.934 

1.08084  4.463 

•3 

.10092  7.616 

1.09096  6.890 

1.08200  6.234 

1.07395  5.640 

1.06671  5.103 

•35 

.07699  8.378 

1.06948  7.581 

I.O627I  6.86O 

1.05661  6.207 

1.05112  5.617 

1.4 

1.05173  8.933 

1.04674  8.085 

1.04224  7.317 

1.03817  6.622 

1.03450  5.993 

i-45 

1.02584  9.272 

1.02338  8.393 

1.02116  7.596 

1.01914  6.875 

I.OI733  6.222 

-5 

i.ooooo  9.385 

i.ooooo  8.496 

i.ooooo  7.689 

i.ooooo  6.960 

i.ooooo  6.299 

•55 

0.97481  9.272 

0.97715  8.393 

0.97928  7.596 

0.98122  6.875 

0.98298  6.222 

.6 

0.95082  8.933 

0-95534  8.085 

0.95947  7.317 

0.96323  6.622 

0.96665  5-993 

•65 

0.92852  8.378 

0.93504  7-58i 

0.94099  6.860 

0.94642  6.207 

0.95137  5-617 

-7 

0.90833  7-616 

0.91663  6.890 

0.92421  6.234 

0.93114  5.640 

0.93746  5.103 

-75 

0.89063  6.666 

0.90047  6.029 

0.90947  5-454 

0.91769  4.934 

0.92521  4-463 

.8 

0-87573  5-549 

0.88684  5.017 

0.89702  4.538 

0.90634  4.104 

0.91485  3.712 

•85 

0.86387  4.291 

0-87599  3.879 

0.88711  3.508 

0.89728  3.172 

0.90659  2.869 

•9 

0.85526  2.924 

0.868  1  r  2.643 

0.87990  2.389 

0.89069  2.161 

0.90057  1.953 

•95 

0.85004  1.481 

0.86333  1.338 

0.87552  I.2IO 

0.88669  1.094 

0.89692  0.989 

2.0 

0.84828  o.oo 

0.86172  o.oo 

0.87405  o.oo 

0.88535  o-oo 

0.89569  o.oo 

Note.     Negative  quantities  are  in  heavy  type. 

Examples,  tanh 

(1.25  +  to.  25) 

=  0.89063  /6°.666 

=  0.89063  /6°.3g 

'•58'. 

tanh  (1.25  +  i  1.25)  =  1.12280  \6°.666  =  1. 12280  \6°.39'.58*. 


[127] 


TABLE  XII.     HYPERBOLIC  TANGENTS,     tanh  (*  +  iq)  =  r  /y.     CONTINUED 


*=  i-S 

x  =  i-5S 

x  —  1.6 

x  =  1.65 

x  =  1.7 

q 

r    7 

r     y 

r    7 

r    7 

r    7 

o 

o 

o 

o 

0 

o 

0.90515  o.oo 

0.91379  o.oo 

0.92167  o.oo 

0.92886  o.oo 

0.93541  o.oo 

0.05 

0.90627  0.894 

0.91481  0.809 

0.92260  0.732 

0.92970  O.662 

0.93618  0.599 

O.I 

0.90960  1.767 

0.91785  1.598 

0.92537  1.446 

0.93223  1.308 

0.93848  1.183 

0.15 

0.91509  2.595 

0.92285  2.347 

0.92993  2.123 

0.93638  I.92I 

0.94226  1.738 

0.2 

0.92263  3.357 

0.92972  3.038 

0.93619  2.748 

0.94207  2.486 

0.94744  2.249 

0.2S 

0.93207  4-038 

0.93832  3-653 

0.94401  3.305 

O.9492O  2.990 

0.95392  2.705 

o-3 

0.94323  4.617 

0.94847  4.177 

0.95325  3-780 

0.95760  3.420 

0.96155  3.095 

o-35 

0.95588  5.083 

0.95998  4.599 

0.96371  4.161 

0.96710  3.766 

0.97018  3.407 

0.4 

0.96976  5.423 

0.97259  4.908 

0.97516  4.440 

0.97749  4.019 

0.97961  3.636 

0-45 

0.98458  5.631 

0.98603  5.096 

0-98735  4-6n 

0.98854  4.173 

0.98962  3.776 

o-S 

i.ooooo  5.700 

i.ooooo  5.159 

i.ooooo  4.668 

i.ooooo  4.225 

i.ooooo  3.822 

o-5S 

1.01566  5.631 

1.01417  5.096 

1.01281  4.611 

1.01159  4-173 

1.01049  3-776 

0.6 

1.03118  5.423 

1.02818  4.908 

1.02548  4.440 

1.02303  4.019 

1.02082  3.636 

0.65 

1.04616  5.083 

1.04169  4.599 

1.03766  4.161 

1.03402  3.766 

1.03074  3.407 

0.7 

1.06019  4.617 

1-05433  4-177 

1.04904  3.780 

1.04428  3.420 

1-03999  3-095 

0-75 

1.07289  4.038 

1.06574  3.653 

1-05931  3-305 

1-05353  2.990 

1.04831  2.705 

0.8 

1.08386  3.357 

1-07559  3-038 

1.06817  2.748 

1.06149  2.486 

1.05548  2.249 

0.85 

1.09279  2.595 

1.08360  2.347 

1-07535  2.123 

1.06795  1-921 

1.06128  1.738 

0.9 

1.09938  1.767 

1.08951  1.598 

1.08065  1.446 

1.07270  1.308 

1.06556  1.183 

o-95 

1.10343  0.894 

1.09313  0.809 

1.08390  0.732 

1.07562  0.662 

1.06817  0.599 

I.O 

1.10479  o.oo 

1.09436  o.oo 

1.08500  o.oo 

1.07659  o.oo 

1.06906  o.oo 

1.05 

1.10343  0.894 

1.09313  0.809 

1.08390  0.732 

1.07562  0.662 

1.06817  0.599 

i.i 

1.09938  1.767 

1.08951  1.598 

1.08065  1.446 

1.07270  1.308 

1.06556  1.183 

i-iS 

1.09279  2.595 

1.08360  2.347 

1-07535  2.123 

1.06795  1-921 

1.06128  1.738 

1.2 

1.08386  3.357 

1-07559  3.038 

1.06817  2.748 

1.06149  2.486 

1.05548  2.249 

1.25 

1.07289  4.038 

1.06574  3-653 

1.05931  3.305 

1.05353  2.990 

1.04831  2.705 

i-3 

1.06019  4.617 

1.05433  4.177 

1.04904  3.780 

1.04428  3.420 

1.03999  3-095 

1-35 

1.04616  5.083 

1.04169  4.599 

1.03766  4.161 

1.03402  3.766 

1.03074  3.407 

1.4 

1.03118  5.423 

1.02818  4.908 

1.02548  4.440 

1.02303  4.019 

1.02082  3.636 

I-4S 

1.01566  5.631 

1.01417  5.096 

1.01281  4.611 

1.01159  4.173 

1.01049  3.776 

i-S 

i.ooooo  5.700 

i.ooooo  5.159 

i.ooooo  4.668 

i.ooooo  4.225 

i.ooooo  3.822 

1-55 

0.98458  5.631 

0.98603  5.096 

0.98735  4.611 

0.98854  4.173 

0.98962  3.776 

1.6 

0.96976  5-423 

0.97259  4.908 

0.97516  4.440 

0.97749  4.019 

0.97961  3.636 

1.65 

0.95588  5.083 

0.95998  4.599 

0.96371  4.161 

0.96710  3.766 

0.97018  3.407 

i-7 

0.94323  4.617 

0.94847  4.177 

0.95325  3.780 

0.95760  3.420 

0.96155  3.095 

i-75 

0.93207  4.038 

0.93832  3.653 

0.94401  3.305 

0.94920  2.990 

0.95392  2.705 

1.8 

0.92263  3-357 

0.92972  3.038 

0.93619  2.748 

0.94207  2.486 

0.94744  2.249 

1.85 

0.91509  2.595 

0.92285  2.347 

0.92993  2.123 

0.93638  1.921 

0.94226  1.738 

1.9 

0.90960  1.767 

0.91785  1.598 

0.92537  1.446 

0.93223  1.308 

0.93848  1.183 

1-95 

0.90627  0.894 

0.91481  0.809 

0.92260  0.732 

0.92970  0.662 

0.93618  0.599 

2.0        0.90515   o.oo        0.91379   o.oo        0.92167   o.oo        0.92886   o.oo        0.93541    o.oo 

Note.  Negative  quantities  are  in  heavy  type. 

Examples,     tanh  (1.7  +  *'  Q-7)  =  1-03999  /3°-°95  =  1-03999  /3°-°5>-42y. 
tanh  (1.6  +  »  u)  =  0.95325  \3°.78o  =  0.95325  \3°46'.48*. 


[128] 


TABLE  XII.     HYPERBOLIC  TANGENTS,     tanh  (*  +  iq)  =  r  fa.     CONTINUED 


=  1.75 


x  =  1.8 


*  =  1.85 


*  =  1.9 


x  =  1.95 


9 

r    y 

r    y 

r    y 

r     y 

r     7 

o 

o 

o 

o 

o 

o 

0.94138  o.oo 

0.94681  o.oo 

0.95175  o.oo 

0.95624  o.oo 

0.9603  2  O.OO 

0.05 

0.94208  0.542 

0.94745  0.490 

0.95232  0.443 

0.95677  0.401 

0.96080  0.363 

O.I 

0.94417  1.070 

0.94935  0.968 

0.95406  0.876 

0.95834  0.793 

O.96222  O.7I7 

0.15 

0.94761  1.572 

0.95247  1.422 

0.95690  1.287 

0.96092  1.165 

0.96457  1.053 

O.2 

0.95232  2.035 

0.95676  1.841 

0.96079  1.666 

0.96445  1.507 

0.96778  1.364 

0.25 

0.95821  2.448 

0.96211  2.215 

0.96565  2.004 

0.96886  1.813 

0.97178  1.640 

°-3 

0.96514  2.800 

0.96840  2.533 

0.97136  2.292 

0.97405  2.074 

0.97649  1.877 

o-3S 

0.97297  3.083 

0.97551  2.789 

0.97781  2.524 

0.97990  2.285 

0.98179  2.067 

0.4 

0.98153  3.291 

0.98327  2.977 

0.98485  2.694 

0.98628  2.438 

0.98757  2.2O6 

o-4S 

0.99061  3.417 

0.99149  3.092 

0.99230  2.798 

0.99303  2.531 

0.99369  2.291 

o-S 

i.  ooooo  3.459 

i.ooooo  3.130 

i.ooooo  2.833 

i.ooooo  2.562 

i.ooooo  2.319 

o-5S 

1.00948  3.417 

1.00858  3.092 

1.00776  2.798 

1.00702  2.531 

1.00635  2.291 

0.6 

1.01882  3.291 

1.01702  2.977 

1.01539  2-694 

1.01392  2.438 

1.01258  2.206 

0.65 

1.02778  3.083 

1.02511  2.789 

1.02269  2.524 

1.02051  2.285 

1.01854  2.067 

0.7 

1.03612  2.800 

1.03263  2.533 

1.02948  2.292 

1.02664  2.074 

1.02408  1.877 

o.7S 

1.04362  2.448 

1.03939  2.215 

1.03558  2.004 

1.03214  1.813 

1.02904  1.640 

0.8 

1.05007  2.035 

1.04520  1.841 

1.04081  1.666 

1.03686  1.507 

1.03329  1.364 

0.85 

1.05529  1.572 

1.04990  1.422 

1.04504  1.287 

1.04067  1.165 

1.03673  1.053 

0.9 

1.05913  1.070 

1.05336  0.968 

1.04816  0.876 

1.04347  0.793 

1.03926  0.717 

0-95 

1.06148  0.542 

1.05547  0.490 

1.05006  0.443 

1.04519  0.401 

1.04080  0.363 

I.O 

1.06228  o.oo 

1.05619  o.oo 

1.05070  o.oo 

1.04576  o.oo 

1.04131  o.oo 

1.05 

1.06148  0.542 

1.05547  0.490 

1.05006  0.443 

1.04519  0.401 

1.04080  0.363 

i.i 

1.05913  1.070 

1.05336  0.968 

1.04816  0.876 

1.04347  0.793 

1.03926  0.717 

i-iS 

1.05529  1.572 

1.04990  1.422 

1.04504  1.287 

1.04067  1.165 

1.03673  1.053 

1.2 

1.05007  2.035 

1.04520  1.841 

1.04081  1.666 

1.03686  1.507 

1.03329  1.364 

1.25 

1.04362  2.448 

I-°3939  2.215 

1.03558  2.004 

1.03214  1.813 

1.02904  1.640 

1-3 

1.03612  2.800 

1.03263  2.533 

1.02948  2.292 

1.02664  2.074 

1.02408  1.877 

i-3S 

1.02778  3.083 

1.02511  2.789 

1.02269  2.524 

1.02051  2.285 

1.01854  2.067 

1.4 

1.01882  3.291 

1.01702  2.977 

1.01539  2.694 

1.01392  2.438 

1.01258  2.206 

1-45 

1.00948  3.417 

1.00858  3.092 

1.00776  2.798 

1.00702  2.531 

1.00635  3-29i 

i-5 

i.ooooo  3.459 

i.ooooo  3.130 

i.ooooo  2.833 

i.ooooo  2.562 

i.ooooo  2.319 

i-S5 

0.99061  3.417 

0.99149  3-092 

0.99230  2.798 

0.99303  2.531 

0.99369  2.291 

1.6 

0.98153  3.291 

0.98327  2.977 

0.98485  2.694 

0.98628  2.438 

0.98757  2.206 

1.65 

0.97297  3.083 

0.97551  2.789 

0.97781  2.524 

0.97990  2.285 

0.98179  2.067 

1-7 

0.96514  2.800 

0.96840  2.533 

0.97136  2.292 

0.97405  2.074 

0.97649  1.877 

J-75 

0.95821  2.448 

0.96211  2.215 

0.96565  2.004 

0.96886  1.813 

0.97178  1.640 

1.8 

0.95232  2.035 

0.95676  1.841 

0.96079  1.666 

0.96445  1.507 

0.96778  1.364 

1.85 

0.94761  1.572 

0.95247  1.422 

0.95690  1.287 

0.96092  1.165 

0.96457  1.053 

1.9 

0.94417  1.070 

0.94935  0.968 

0.95406  0.876 

0.95834  0.793 

0.96222  0.717 

1-95 

0.94208  0.542 

0.94745  0.490 

0.95232  0.443 

0.95677  0.401 

0.96080  0.363 

2.O 

0.94138  o.oo 

0.94681  o.oo 

o-9S  1  75  °-°° 

0.95624  o.oo 

0.96032  o.oo 

Note.     Negative  quantities  are  in  heavy  type. 

Examples,  tanh 

(1.9  +  i  0.05)  = 

0.95677  /o°.40i 

=  0.95677  /o°.24' 

,04'. 

tanh  (i  95  +  i . 


i.ooo\2°.i9'.o8*. 


[129] 


TABLE  XII.     HYPERBOLIC  TANGENTS,     tanh  (*  +  iq)  -  f 


CONTINUED 


x  =  2.05 


*  =  2.15 


0 

o.os 

O.I 

0.15 

O.2 

0.96403  o.oo 
0.96446  0.3  28 
0.96576  0.649 

0.96789  0.953 
0.97080  1.234 

0.96740  o.oo 
0.96779  0.297 
0.96897  0.587 
0.97090  0.862 

0.97354  1.117 

0.97045  o.oo 
0.97081  0.268 

0.97188  0.531 
0.97363  0.780 
0.97603  i.oio 

0.97323  o.oo 

0.97355  0.243 
0.97452  0.481 
0.97611  0.706 
0.97829  0.914 

0.97574  o.oo 

0.97604  O.22O 

0.97692  0.435 
0.97842  0.639 
0.98033  0.827 

0.25 
0-3 
0-35 
0.4 

0-45 

0.97443  1.484 
0.97870  1.698 
0.98351  1.870 
0.98875  1.996 

0.99429  2.072 

0.97684  1.343 
0.98071  1.537 

0.98507  1.692 
0.98981  1.  806 
0.99483  1.876 

0.97902  1.215 
0.98253  1.390 
0.98648  1.531 
0.99078  1.634 
0.99532  1.698 

0.98100  1.099 
0.98418  1.258 
0.98776  1.385 
0.99165  1.479 
0.99577  1-536 

0.98279  0.995 
0.98567  1.138 

0.98892  1.253 

0.99244  1.337 
0.99617  1.389 

0-5 
0-55 

0.6 
0.65 
0.7 

i.ooooo  2.098 
1.00574  2.072 
1.01138  1.996 
1.01676  1.870 
1.02176  1.698 

i.ooooo  1.899 
1.00520  1.876 
1.01029  1.  806 
1.01516  1.692 

1.01967  1.537 

i.ooooo  1.718 
1.00470  1.698 
1.00931  1.634 
1.01371  1.531 
1.01778  1.390 

i.ooooo  1.555 
1.00425  1.536 
1.00842  1.479 
1.01240  1.385 
1.01608  1.258 

i.ooooo  1.406 

1.00385  1.389 
1.00762  1.337 

I.OII2I  1.253 

1.01454  1.138 

0-75 
0.8 
0.85 
0.9 
0-95 

1.02624  1.484 
1.03008  1.234 

1.03318  0.953 
1.03545  0.649 
1.03685  0.328 

1.02371  1.343 
1.02718  1.117 
1.02998  0.862 
1.03203  0.587 
1.03328  0.297 

1.02143  1-215 
1.02456  i.oio 
1.02708  0.780 

1.02894  0.531 

1.03007  0.268 

1.01937  1.099 

I.O222O  0.914 

1.02447  0.706 
1.02614  0.481 
1.02717  0.243 

1.01751  0.995 

1.02006  0.827 
1.02206  0.639 
1.02363  0.435 

I.O2455  O.22O 

I.O 

1.05 
i.i 

1.2 

1.03731  o.oo 
1.03685  0.328 

1.03545  0.649 
1.03318  0.953 
1.03008  1.234 

1.03370  o.oo 
1.03328  0.297 
1.03202  0.587 
1.02998  0.862 
1.02718  1.117 

1.03045  o.oo 
1.03007  0.268 
1.02894  0.531 
1.02708  0.780 
1.02456  i.oio 

1.02751  o.oo 
1.02717  0.243 
1.02614  0.481 
1.02447  0.706 

I.O222O  O.9I4 

1.02486  o.oo 

1.02455  0.220 

1.02363  0.435 

1.02206  0.639 
1.02006  0.827 

1.25 

1-35 
1.4 
1-45 

1.02624  1.484 
1.02176  1.698 
1.01676  1.870 
1.01138  1.996 
1.00574  2.072 

1.02371  1.343 

1.01967  1.537 
1.01516  1.692 
1.01029  1-806 
1.00520  1.876 

1.02143  1-215 
1.01778  1.390 
1.01371  1.531 

1.00931  1.634 
1.00470  1.698 

1.01937  1.099 

1.01608  1.258 
1.01240  1.385 
1.00842  1.479 
1.00425  1.536 

1.01751  0.995 
1.01454  1.138 

I.OII2I  1.253 

1.00762  1.337 

1.00385  1.389 

i-S 
i-SS 
1.6 

1.65 
i-7 

i.ooooo  2.098 
0.99429  2.072 

0.98875  1.996 
0.98351  1.870 
0.97870  1.698 

i.ooooo  1.899 

0.99483  1.876 
0.98981  1.  806 
0.98507  1.692 

0.98071  1.537 

i.ooooo  1.718 
0.99532  1.698 
0.99078  1.634 
0.98648  1.531 
0.98253  1.390 

i.ooooo  1.555 

0.99577  1-536 
0.99165  1.479 
0.98776  1.385 
0.98418  1.258 

i.ooooo  1.406 
0.99617  1.389 
0.99244  1.337 
0.98892  1.253 
0.98567  1.138 

i.'s 

1.85 
1.9 
1-95 

0.97443  1.484 

0.97080  1.234 

0.96789  0.953 
0.96576  0.649 
0.96446  0.328 

0.97684  1.343 
0.97354  1.  117 

0.97090  0.862 
0.96897  0.587 

0.96779  0.297 

0.97902  1.215 
0.97603  i.oio 

0.97363  0.780 
0.97188  0.531 
0.97081  0.268 

0.98100  1.099 
0.97829  0.914 
0.97611  0.706 

0.97452  0.481 
0.97355  0.243 

0.98279  0.995 
0.98033  0.827 

0.97842  0.639 

0.97692  0.435 

0.97604  O.22O 

2.0 

0.96403  o.oo 

0.96740  o.oo 

0.97045  o.oo 

0.97323  o.oo 

0.97574  o.oo 

Note.  Negative  quantities  are  in  heavy  type. 

Examples,    tanh  (2.2  +  40)  =0.97574  /o°. 


tanh  (2.2  +  i  1.95)  =  0.97604  \o°.22o  =  0.97604  \o°.i3'.i2*. 


[130] 


TABLE  XII.     HYPERBOLIC  TANGENTS,     tanh  (*  +  iq)  =  r  /y.     CONTINUED 


x  =  2.25 

x  =  2.3 

X  =  2.35 

x  =  2.4 

x  =  2.45 

1 

r     y 

*     y 

r     y 

r     y 

r     y 

o 

o 

o 

o 

0 

o 

0.97803  o.oo 

0.98010  o.oo 

0.98197  o.oo 

0.98367  o.oo 

0.98522  o.oo 

0.05 

0.97829  0.199 

0.98034  0.180 

0.98219  0.163 

0.98388  0.147 

0.98540  0.134 

O.I 

0.97909  0.394 

0.98106  0.356 

0.98285  O.322 

0.98447  0.291 

0.98594  0.264 

0.15 

0.98040  0.578 

0.98225  0.523 

0.98393  0.473 

0.98544  0.428 

0.98682  0.388 

0.2 

0.98219  0.748 

0.98387  0.677 

0.98539  0.6l3 

0.98677  0.554 

0.98802  0.501 

0.25 

0.98441  0.900 

0.98589  0.814 

0.98722  0-737 

0.98843  0.666 

0.98953  0.603 

0-3 

0.98703  1.030 

0.98825  0.932 

0.98937  0.843 

0.99037  0.763 

0.99129  0.690 

o-3S 

0.98997  1.134 

0.99092  1.027 

0.99178  0.929 

0.99256  0.841 

0.99326  0.760 

0-4 

0.99316  1.  211 

0.99381  1.095 

0.99440  0.991 

0.99493  0.896 

0.99541  0.812 

o.4S 

0.99653  1.257 

0.99686  1.138 

0.99716  I.O3O 

0.99743  0.931 

0.99767  0.843 

0-5 

i.ooooo  1.273 

i.ooooo  1.152 

i.ooooo  1.042 

i.ooooo  0.942 

i.ooooo  0.853 

o-5S 

1.00348  1.257 

1.00315  1.138 

1.00285  1-030 

1.00258  0.931 

1.00233  0-843 

0.6 

1.00689  I-2II 

1.00623  I-°95 

1.00564  0.991 

1.00510  0.896 

1.00461  0.812 

0.65 

I.OIOI4  I.I34 

1.00917  1.027 

1.00829  0.929 

1.00750  0.841 

1.00678  0.760 

0.7 

I.OI3I4  1.030 

1.01189  0.932 

1.01075  0.843 

1.00972  0.763 

1.00879  0.690 

0-75 

1.01583  O.gOO 

1.01431  0.814 

1.01295  0.737 

1.01171  0.666 

1.01058  0.603 

0.8 

I.Ol8l4  0.748 

1.01640  0.677 

1.01482  0.613 

1.01340  0.554 

I.OI2I2  O.5OI 

0.85 

I.OI999  0.578 

1.01807  0.523 

1.01634  0.473 

1.01477  0.428 

1.01336  0.388 

0.9 

I.02I36  0.394 

1.01930  0.356 

1.01745  °-322 

1.01578  0.291 

I.OI426  0.264 

0-95 

I.O22I9  0.199 

1.02005  0.180 

1.01813  0.163 

1.01639  0.147 

I.OI482  0.134 

I.O 

I.O2247  O-°O 

1.02031  o.oo 

1.01836  o.oo 

1.01659  o.oo 

1.01500  o.oo 

1.05 

I.O22I9  O.IQ9 

1.02005  0.180 

1.01813  0.163 

1.01639  0.147 

1.01482  0.134 

i.i 

I.O2I36  O.394 

1.01930  0.356 

1.01745  0.322 

1.01578  0.291 

1.01426  0.264 

i-iS 

1.01999  O.578 

1.01807  0.523 

1.01634  0.473 

1.01477  0.428 

1.01336  0.388 

1.2 

I.Ol8l4  0.748 

1.01640  0.677 

1.01482  0.613 

1.01340  0.554 

1  .01  21  2  O.5OZ 

•25 

1.01583  O.90O 

1.01431  0.814 

1.01295  0.737 

1.01171  0.666 

I.OIO58  O.6O3 

•3 

I.OI3I4  I.03O 

1.01189  0.932 

1.01075  0.843 

1.00972  0.763 

1.00879  0.60X) 

•35 

I.OIOI4  I-I34 

1.00917   .027 

1.00829  0.929 

1.00750  0.841 

1.00678  0.760 

•4 

1.00689  I  -2  1  1 

1.00623  .095 

1.00564  0.991 

1.00510  0.896 

1.00461  0.812 

•45 

J.00348  1.257 

1.00315  .138 

1.00285  1.030 

1.00258  0.931 

1.00233  0.843 

•5 

i.ooooo  1.273 

i.ooooo  .152 

i.ooooo  1.042 

i.ooooo  0.942 

i.ooooo  0.853 

•55 

0.99653  1.257 

0.99686   .138 

0.99716  1.030 

0.09743  °-93i 

0.99767  0.843 

1.6 

0.99316  Z.2II 

0.99381  .095 

0.99440  0.991 

0.99493  0.896 

0.99541  0.812 

1.65 

0.98997  1.  134 

0.99092  1.027 

0.99178  0.929 

0.99256  0.841 

0.99326  0.760 

i-7 

0.98703   1.030 

0.98825  0.932 

0.98937  0.843 

0.99037  0.763 

0.99129  0.690 

1-75 

0.98441  O.9OO 

0.98589  0.814 

0.98722  0.737 

0.98843  0.666 

0.98953  0.603 

1.8 

0.98219  0.748 

0.98387  0.677 

0.98539  0.613 

0.98677  0.554 

0.98802  0.501 

1.85 

0.98040  O.578 

0.98225  0.523 

0.98393  0.473 

0.98544  0.428 

0.98682  0.388 

1.9 

0.97909  0.394 

0.98106  0.356 

0.98285  0.322 

0.98447  0.291 

0.98594  0.164 

1-95 

0.97829  0.199 

0.98034  0.180 

0.98219  0.163 

0.98388  0.147 

0.98540  0.134 

*.o        0.97803   o.oo        0.98010   o.oo        0.98197   o.oo        0.98367   o.oo        0.98522   o.oo 

Note.  Negative  quantities  are  in  heavy  type. 

Examples,     tanh  (2.45  +  »  oj)  =  1.00879  /o°-69Q  =  1.00879  /o°.4if-24'- 
tanh  (2.45  +  *  rj)  =  0.99129  \o°.6oo  =  0.99129  \o°.4i'-24r. 


[131] 


TABLE  XII.     HYPERBOLIC  TANGENTS,     tanh  (x  +  iq)  =  r  fa.     CONTINUED 


*  =  2.5 

x  =  2.55 

X  =  2.6 

*  =  2.65 

x  =  2.7 

9 

r     y 

f      7 

r     7 

r     7 

f     Y 

0 

o 

0 

o 

0 

o 

0.98661  o.oo 

0.98788  o.oo 

0.98903  o.oo 

0.99007  o.oo 

0.99101  o.oo 

0.05 

0.98678  O.I2I 

0.98803  0.109 

0.98916  0.099 

0.99019  0.089 

0.99112  0.081 

O.I 

0.98727  0.239 

0.98847  0.216 

0.98956  0.195 

0.99055  0.177 

0.99145  0.160 

0.15 

0.98806  0.350 

0.98919  0.317 

0.99022  0.287 

0.99114  0.260 

0.99198  0.235 

O.2 

0.98916  0.454 

0.99018  0.411 

0.99111  0.372 

0.99196  0.336 

0.99272  0.304 

0.2$ 

0.99052  0.546 

0.99142  0.494 

0.99223  0.447 

0.99297  0.405 

0-99363  0-366 

0-3 

O.992II  O.625 

0.99286  0.565 

0.99354  0.512 

0.99415  0.463 

0.99471  0.419 

0-35 

0.99390  0.688 

0.99448  0.623 

0.99500  0.563 

0.99548  0.510 

0.99591  0.461 

0.4 

0.99584  0.734 

0.99624  0.665 

0.99660  0.601 

0.99692  0.544 

0.99721  0.492 

o-4S 

0.99789  0.763 

0.99809  0.690 

0.99828  0.624 

0.99844  0.565 

0.99859  0.511 

o.S 

i.ooooo  0.772 

i.ooooo  0.699 

i.ooooo  0.632 

i.ooooo  0.573 

i.ooooo  0.518 

0-55 

I.OO2II  0.763 

1.00191  0.690 

1.00173  0.624 

1.00156  0.565 

1.00141  0.511 

0.6 

I.004I7  0.734 

1.00377  0.665 

1.00342  0.601 

1.00309  0.544 

1.00280  0.492 

0.65 

1.00614  0.688 

1.00555  0-623 

1.00502  0.563 

1.00454  0.510 

1.00411  0.461 

0.7 

1.00795  0.625 

1.00719  0.565 

1.00651  0.512 

1.00589  0.463 

1.00532  0.419 

0-75 

1.00958  0.546 

1.00866  0.494 

1.00783  0.447 

1.00708  0.405 

1.00641  0.366 

0.8 

1.01096  0.454 

1.00991  0.411 

1.00897  0.372 

1.00811  0.336 

1.00733  0.304 

0.85 

1.01208  0.350 

1.01092  0.317 

1.00988  0.287 

1.00894  0.260 

1.00808  0.235 

0.9 

1.01290  0.239 

1.01166  0.216 

1.01055  0.195 

1.00954  0.177 

1.00863  o.i  60 

o-9S 

1.01340  o.i  21 

I.OI2I2  O.IO9 

1.01096  0.099 

1.00991  0.089 

1.00896  0.081 

I.O 

1.01357  0.00 

1.01227  o.oo 

I.  OHIO  O.OO 

1.01003  O'O° 

1.00907  o.oo 

i.  05 

I.OI340  O.I2I 

I.OI2I2  O.IO9 

1.01096  0.099 

1.00991  0.089 

1.00896  0.081 

i.i 

1.01290  0.239 

1.01166  0.216 

1.01055  0-195 

1.00954  0.177 

1.00863  o.i  60 

i.iS 

1.01208  0.350 

I.OIO92  O.3I7 

1.00988  0.287 

1.00894  0.260 

1.00808  0.235 

1.2 

1.01096  0.454 

I.OO99I  O.4II 

1.00897  0.372 

1.  0081  1  0.336 

1.00733  0.304 

I.2S 

1.00958  0.546 

I.OO866  0.494 

1.00783  0.447 

1.00708  0.405 

1.00641  0.366 

1-3 

1.00795  0.625 

I.007I9  0.565 

1.00651  0.512 

1.00589  0.463 

1.00532  0.419 

1-35 

1.00614  0.688 

!-oo555  0.623 

1.00502  0.563 

1.00454  0.510 

1.00411  0.461 

1.4 

1.00417  0.734 

1.00377  0.665 

1.00342  0.601 

1.00309  0.544 

1.00280  0.492 

i-45 

I.OO2II  0.763 

1.00191  0.690 

1.00173  0.624 

1.00156  0.565 

1.00141  0.511 

i-5 

i.ooooo  0.772 

i.ooooo  0.699 

i.ooooo  0.632 

i.ooooo  0.573 

i.ooooo  0.518 

i-SS 

0.99789  0.763 

0.99809  0.690 

0.99828  0.624 

0.99844  0.565 

0.99859  0.511 

1.6 

0.99584  0.734 

0.99624  0.665 

0.99660  0.601 

0.99692  0.544 

0.99721  0.492 

1.65 

0.99390  0.688 

0.99448  0.623 

0.99500  0.563 

0.99548  0.510 

0.99591  0.461 

1-7 

0.99211  0.625 

0.99286  0.565 

0.99354  0.512 

0.99415  0.463 

0.99471  0.419 

1-75 

0.99052  0.546 

0.99142  0.494 

0.99223  0.447 

0.99297  0.405 

0.99363  0.366 

1.8 

0.98916  0.454 

0.99018  0.411 

0.99111  0.372 

0.99196  0.336 

0.99272  0.304 

1.85 

0.98806  0.350 

0.98919  0.317 

0.99022  0.287 

0.99114  0.260 

0.99198  0.235 

1.9 

0.98727  0.239 

0.98847  0.216 

0.98956  0.195 

0.99055  0.177 

0.99145  0.160 

i-95 

0.98678  O.I2I 

0.98803  0.109 

0.98916  0.099 

0.99019  0.089 

0.99112  0.081 

2.0        0.98661    o.oo        0.98788   o.oo        0.98903   o.oo        0.99007   o.oo        0.99101    o.oo 

Note.  Negative  quantities  are  in  heavy  type. 

Examples,    tanh  (2.5  +  *  0.25)  =  0^99052  /o°.546  =  0.99052  /  o°  . 


tanh  (2.5  +  *  1.75)  =  0.99052  \o°.546  =  0.99052  \o°.32'.46*. 


TABLE  XII.     HYPERBOLIC  TANGENTS,     tanh  (x  +  iq)  -  r^y.     CONTINUED 


=  2.75 


=  2.8 


*  =  2.85 


x  =  2.9 


x  =  2.95 


o 
0.05 

O.I 

0.15 

O.2 

0.99186  o.oo 
0.99196  0.073 
0.99226  0.145 
0.99275  0.213 
0.99341  0.275 

0.99263  o.oo 
0.99272  0.066 
0.99299  0.131 
0.99343  0.192. 
0.99404  0.249 

Q-99333  o.oo 
0.99341  0.060 
0.99366  0.119 
0.99406  0.174 
0.99460  0.226 

0.99396  o.oo 
0.99404  0.054 
0.99426  0.107 
0.99462  0.158 
0.99511  0.204 

0.99454  o.oo 
0.99460  0.049 
0.99480  0.097 
0.99513  0.143 
0-99558  0.184 

O.2S 

°-3 

0.35 
0.4 

0.45 

0.99424  0.331 
0.99521  0.379 
0.99630  0.417 
0.99748  0.446 
0.99872  0.463 

0.99478  0.300 
0.99566  0.343 
0.99665  0.378 
0.99772  0.403 
0.99884  0.419 

0.99528  0.271 
0.99607  0.310 
0.99697  0.342 
0.99793  0.365 
0.99895  0.379 

0-99573  0.245 
0.99645  0.281 
0.99726  0.309 
0.99813  0.330 
0.99905  0.343 

0.99613  O.222 
0.99679  0.254 
0.99752  O.28O 
0.99831  0.299 
0.99914  0.310 

o-5 
o.SS 
0.6 
0.65 
0.7 

i  .00000  0.468 
1.00128  0.463 
1.00253  0-446 
1.00372  0.417 
1.00482  0.379 

i  .00000  0.424 
1.00116  0.419 
1.00229  °-4°3 
1.00336  0.378 
1.00436  0.343 

i  .00000  0.383 
1.00105  0.379 
1.00207  0.365 
1.00304  0.342 
1.00394  0.310 

i  .00000  0.347 
1.00095  0.343 
1.00187  0.330 
1.00275  0.309 
1.00356  0.281 

i.  ooooo  0.313 
1.00086  0.310 
1.00169  0.209 
1.00249  0.280 
1.00323  0.254 

0-75 
0.8 
0.85 
0.9 
o-95 

1.00580  0.331 
1.00664  0.275 
1.00731  0.213 
1.00780  0.145 
1.00810  0.073 

1.00524  0.300 
1.00600  0.249 
i  .0066  1  0.192 
1.00706  0.131 
1.00733  0.066 

1.00474  0.271 
1.00543  0.226 
1.00598  0.174 
1.00638  0.119 
1.00663  0.060 

1.00429  0.245 
1.00491  0.204 
1.00541  0.158 
1.00578  0.107 
1.00600  0.054 

1.00388  O.222 
1.00444  0.184 
1.00489  O.I43 
I.OO522  O.O97 
1.00543  0.049 

I.O 

1.05 
I.I 

1-15 

1.2 

1.00821  o.oo 
1.00810  0.073 
1.00780  0.145 
1.00731  0.213 
1.00664  0.275 

1.00742  o.oo 
1.00733  0.066 
1.00706  0.131 
1.00661  0.192 
1.00600  0.249 

1.00671  o.oo 
1.00663  0.060 
1.00638  0.119 
1.00598  0.174 
1.00543  0.226 

1.00607  o.oo 
1.00600  0.054 
1.00578  0.107 
1.00541  0.158 
1.00491  0.204 

1.00549  o.oo 

1.00543  0.049 
I.OO522  O.O97 
1.00489  O.I43 
I.OO444  0.184 

1.25 

i-3 
1-35 
1.4 

1-45 

1.00580  0.331 
1.00482  0.379 
1.00372  0.417 
1.00253  0-446 
1.00128  0.463 

1.00524  0.300 
1.00436  0.343 
1.00336  0.378 
1.00229  0.403 
1.00116  0.419 

1.00474  0.271 
1.00394  0.310 
1.00304  0.342 
1.00207  0.365 
1.00105  0.379 

1.00429  0.245 
1.00356  0.281 
1.00275  0.309 
1.00187  0.330 
1.00095  0.343 

1.00388  O.222 
1.00323  0.254 
I.OO249  O.28O 
I.OOl69  O.299 
1.00086  0.310 

•5 
•55 
.6 

-65 
•7 

1  .00000  0.468 
0.99872  0.463 
0.99748  0.446 

0.99630  0.417 
0.99521  0.379 

i  .00000  0.424 
0.99884  0.419 
0.99772  0.403 
0.99665  0.378 
0.99566  0.343 

1  .00000  0.383 
0.99895  0.379 
0.99793  0.365 
0.99697  0.342 
0.99607  0.310 

1  .00000  0.347 

0.99905  0.343 
0.99813  0.330 
0.99726  0.309 
0.99645  0.281 

i  .ooooo  0.313 
0.99914  0.310 
0.99831  0.299 
0.99752  0.280 

0.99679  0.254 

•75 
.8 

.85 
•9 
•95 

0.99424  0.331 
0.99341  0.275 
0.99275  0.213 

0.99226  0.145 
0.99196  0.073 

0.99478  0.300 
0.99404  0.249 
0.99343  0.192 
0.99299  0.131 
0.99272  0.066 

0.99528  0.271 
0.99460  0.226 
0.99406  0.174 
0.99366  0.119 
0.99341  0.060 

0-99573  0.245 
0.99511  0.204 
0.99462  0.158 
0.99426  0.107 
0.99404  0.054 

0.99613  0.222 
0.99558  0.184 
0.99513  0.143 
0.99480  O.O97 
0.99460  O.O49 

2.0 

0.99186  o.oo 

0.99263  o.oo 

0-99333  o-°o 

0.99396  o.oo 

0.99454  0.00 

Note.     Negative  quantities  are  in  heavy  type. 

Examples,  tanh  (2.9  +  i  0.5)  = 

i.oooo/o°.347  = 

i.oooo  /o°.26'.49ir. 

tanh  (2.95  +  *  1.75)  =  0.99613  \o°.222  =  0.99613  \o°.i3'.i9'. 


TABLE  XII.     HYPERBOLIC  TANGENTS,     tanh  (x  +  iq)  =  r 


CONTINUED 


X  =  . 

5.0 

x  =  3 

•05 

*  =  . 

J-i 

*  =  3 

•15 

*  =  . 

5-2 

9 

r 

7 

r 

y 

r 

7 

r 

7 

f 

7 

0 

o 

o 

o 

o 

0 

0-9950S 

o.oo 

0.99552 

O.OO 

0-99595 

O.OO 

0.99633 

O.OO 

0.99668 

O.OO 

0.05 

0.99511 

0.044 

0.99558 

O.O4O 

0.99600 

0.037 

0.99638 

0.033 

0.99672 

0.030 

O.I 

0.99530 

0.088 

0-99575 

0.079 

0.99615 

O.O72 

0.99651 

0.065 

0.99685 

0.059 

0.15 

0-99559 

0.129 

0.99601 

O.II7 

0.99639 

0.106 

0.99673 

0.096 

0.99704 

0.086 

O.2 

0.99600 

0.167 

0.99638 

O.ISI 

0.99672 

0.137 

0.99703 

0.124 

0.99732 

0.112 

0.25 

0.99650 

O.2OI 

0.99683 

0.182 

0.99714 

0.165 

0.99741 

0.149 

0.99765 

0.135 

o-3 

0.99709 

0.229 

0-99737 

O.2O8 

0.99762 

o.i  88 

0.99784 

0.170 

0.99805 

0.154 

o-3S 

0-99775 

0.253 

0.99797 

O.229 

0.99816 

0.207 

0.99833 

o.i  88 

0.99849 

0.170 

0.4 

0.99847 

O.27O 

0.99862 

0.244 

0.99875 

O.22I 

0.99887 

O.2OO 

0.99897 

0.181 

0-45 

o.99923 

0.281 

0.99930 

0.254 

0-99937 

0.230 

0-99943 

O.2O8 

0.99948 

o.i  88 

o-5 

I.OOOOO 

0.284 

I.OOOOO 

0.257 

I.OOOOO 

0.233 

I.OOOOO 

O.2II 

I.OOOOO 

0.191 

0-55 

1.00078 

0.28l 

1.00070 

0.254 

1.00064 

0.230 

1.00058 

O.2O8 

1.00052 

o.i  88 

0.6 

1.00153 

0.27O 

1.00139 

0.244 

1.00125 

0.221 

1.00114 

O.2OO 

1.00103 

0.181 

0.65 

1.00225 

0.253 

1.00204 

O.229 

1.00185 

O.2O7 

1.00167 

o.i  88 

1.00151 

0.170 

0.7 

1.00292 

O.229 

I  00264 

0.208 

1.00239 

o.i  88 

1.00216 

0.170 

1.00196 

0.154 

0-75 

1.00351 

O.2OI 

1.00318 

0.182 

1.00287 

0.165 

1.00260 

0.149 

1.00235 

0.135 

0.8 

1.00402 

0.167 

1.00363 

O.I5I 

1.00329 

0.137 

1.00297 

0.124 

1.00269 

O.I  1  2 

0.85 

1.00443 

O.I29 

1.00400 

O.II7 

1.00360 

o.i  06 

1.00328 

0.096 

1.00297 

0.086 

0.9 

1.00473 

0.088 

1.00427 

0.079 

1.00387 

0.072 

1.00350 

0.065 

1.00316 

0.059 

o-9S 

1.00491 

O.O44 

1.00444 

O.O4O 

1.00399 

0.037 

1.00363 

0.033 

1.00329 

0.030 

.0 

1.00497 

O.OO 

1.00450 

O.OO 

1.00407 

O.OO 

1.00368 

O.OO 

1.00333 

O.OO 

•05 

1.00491 

0.044 

1  .00444 

0.040 

1.00399 

0.037 

1.00363 

0.033 

1.00329 

0.030 

.1 

1.00473 

0.088 

1.00427 

0.079 

1.00387 

0.072 

1.00350 

0.065 

1.00316 

0.059 

•15 

1.00443 

0.129 

1.00400 

0.117 

1.00360 

o.i  06 

1.00328 

0.096 

1.00297 

0.086 

.2 

1.00402 

0.167 

1.00363 

0.151 

1.00329 

0.137 

1.00297 

0.124 

1.00269 

O.II2 

•25 

1.00351 

0.201 

1.00318 

0.182 

1.00287 

0.165 

1.00260 

0.149 

1.00235 

0.135 

•3 

1.00292 

O.229 

1.00264 

0.208 

1.00239 

0.188 

1.00216 

0.170 

1.00196 

0.154 

•35 

1.00225 

0.253 

1.00204 

0.229 

1.00185 

0.207 

1.00167 

0.188 

1.00151 

O.I7O 

•4 

1.00153 

O.27O 

1.00139 

0.244 

1.00125 

O.22I 

1.00114 

O.2OO 

1.00103 

O.lSl 

1-45 

1.00078 

0.281 

1.00070 

0.254 

1.00064 

O.23O 

1.00058 

O.2O8 

1.00052 

o.i  88 

i-5 

I.OOOOO 

0.284 

I.OOOOO 

0.257 

I.OOOOO 

0.233 

I.OOOOO 

O.2II 

I.OOOOO 

0.191 

i-55 

0.99923 

0.28l 

0.99930 

0.254 

0-99937 

0.230 

0.99943 

O.2OS 

0.99948 

0.188 

1.6 

0.99847 

O.27O 

0.99862 

0.244 

0.99875 

O.22I 

0.99887 

O.200 

0.99897 

0.181 

1.65 

0.99775 

0.253 

0.99797 

0.229 

0.99816 

O.2O7 

0.99833 

0.188 

0.99849 

0.170 

i-7 

0.99709 

O.220 

0.99737 

0.208 

0.99762 

0.188 

0.99784 

0.170 

0.99805 

o.i54 

1-75 

0.99650 

O.2OI 

0.99683 

0.182 

0.99714 

0.165 

0.99741 

0.149 

0.99765 

0-I3S 

1.8 

0.99600 

O.l67 

0.99638 

0.151 

0.99672 

0.137 

0.99703 

0.124 

0.99732 

O.II2 

1.85 

0.99559 

O.I29 

0.99601 

0.117 

0.99639 

0.106 

0.99673 

0.096 

0.99704 

0.086 

1.9 

0.99530 

0.088 

0-99575 

0.079 

0.99615 

0.072 

0.99651 

0.065 

0.99685 

0.059 

1-95 

0.99511 

0.044 

0.99558 

0.040 

0.99600 

0.037 

0.99638 

0.033 

0.99672 

O.O3O 

2.0 

0.99505 

O.OO 

0.99552 

O.OO 

0-99595 

O.OO 

0.99633 

O.OO 

0.99668 

O.OO 

Note.  Negative  quantities  are  in  heavy  type. 

Examples,     tanh  (3.2  +  10)  =  0.99668  /o°. 

tanh  (3.2  +  i  1.05)  =  1.00329  \o°.o3o  =  1.00329  \o°.i'.48". 


[134] 


TABLE  XII.     HYPERBOLIC  TANGENTS,    tanh  (x  +  iq)  =  r  %     CONTINUED 


=  3-25 


=  3-35 


*  =  3-4 


*  =  3-45 


o 

0.99700  o.oo 

0.99728  o.oo 

0.99754  o.oo 

0.99777  o.oo 

0.99799  o.oo 

0.05 

0.99703  0.027 

0.99732  0.024 

0.99757  0.022 

0.99780  O.O2O 

0.99801  0.018 

O.I 

0.99714  0.053 

0.99741  0.048 

0.99766  0.044 

0.99788  0.039 

0.99809  0.036 

0.15 

0.99732  0.078 

0.99758  0.071 

0.99781  0.064 

0.99802  0.058 

0.99821  0.052 

O.2 

0-99757  O.IOI 

0.99780  0.092 

0.99801  0.083 

0.99820  0.075 

0.99837  0.068 

0.25 

0.99788  0.122 

0.99807  o.iio 

0.99826  0.100 

0.99843  0.090 

0.99858  0.082 

0-3 

0.99823  0.139 

0.99840  0.126 

0.99855  0.114 

0.99869  0.103 

0.99882  0.094 

0-35 

0.99864  0.153 

0.98877  0.139 

0.99888  0.126 

0.99899  0.114 

0.99909  0.103 

0.4 

0.99907  0.164 

0.99916  0.148 

0.99924  0.134 

0.99931  O.I  21 

0.99938  o.iio 

0-45 

0-99953  0.170 

0.99958  0.154 

0.99962  0.139 

0.99965  0.126 

0.99968  0.114 

0-5 

i.ooooo  0.172 

i.ooooo  0.156 

i.ooooo  0.141 

i.ooooo  0.128 

i.ooooo  0.116 

0-55 

1.00047  0.170 

1.00042  0.154 

1.00039  0.139 

1.00035  0.126 

1.00032  0.114 

0.6 

1.00093  0.164 

1.00084  0.148 

1.00076  0.134 

1.00069  O.I  21 

1.00062  o.iio 

0.65 

1.00137  0.153 

1.00124  °-I39 

I.OOII2  0.126 

i.ooioi  0.114 

1.00092  0.103 

0.7 

1.00177  0.139 

1.00160  0.126 

1.00145  0.114 

1.00131  0.103 

1.00118  0.094 

0-75 

I.OO2I3  O.I22 

1.00193  o.iio 

1.00174  0.100 

1.00158  0.090 

1.00143  0.082 

0.8 

1.00244  o.ioi 

I.OO22O  0.092 

1.00199  0.083 

1.00180  0.075 

1.00163  0.068 

0.85 

1.00268  0.078 

1.00243  0.071 

I.OO22O  0.064 

1.00199  0.058 

1.00180  0.052 

0.9 

I.O0286  0.053 

1.00259  0.048 

I.OO234  O.O44 

I.OO2I2  0.039 

1.00192  0.036 

0-95 

I.O0297  O.O27 

1.00269  0.024 

I.OO243  O.O22 

I.OO22O  O.O2O 

1.00199  0.018 

I.O 

1.00301  o.oo 

I.OO273  °'O° 

1.00246  o.oo 

1.00223  o.oo 

1.00202  O.OO 

1.05 

I.OO297  0.027 

1.00269  0.024 

I.OO243  O.O22 

I.OO22O  O.O2O 

1.00199  0.018 

.1 

1.00286  0.053 

1.00259  0.048 

I.OO234  O.O44 

I.OO2I2  0.039 

1.00192  0.036 

•15 

I.OO268  O.O78 

1.00243  0.071 

I.OO22O  O.OO4 

I.OOI99  O.O58 

1.00180  0.052 

.2 

1.00244  o.ioi 

1.00220  O.O92 

I.OOI99  0.083 

I.OOlSo  O.O75 

1.00163  .0.068 

•25 

I.OO2I3  O.I22 

1.00193  o.iio 

1.00174  o.ioo 

I.OOI58  O.O9O 

1.00143  0.082 

•3 

I.OOI77  O.I39 

1.00160  0.126 

I.OOI45  O.II4 

I.OOI3I  O.IO3 

1.00118  0.094 

•35 

I.OOI37  0.153 

1.00124  0.139 

I.OOII2  O.I26 

i.opioi  0.114 

1.00092  0.103 

1.4 

I.OOO93  0.164 

1.00084  0.148 

1.00076  O.I34 

I.OOO69  O.I2I 

1.00062  o.iio 

i-45 

I.OOO47  O.I7O 

1.00042  0.154 

1.00039  °-I39 

I.OOO35  O.I26 

1.00032  0.114 

•5 

i.ooooo  0.172 

i.ooooo  0.156 

i.ooooo  0.141 

i.ooooo  0.128 

i.ooooo  0.116 

•55 

0.99953  0.170 

0.99958  0.154 

0.99962  0.139 

0.99965  O.I26 

0.99968  0.114 

.6 

0.99907  0.164 

0.99916  0.148 

0.99924  0.134 

0.99931  O.I2I 

0.99938  o.iio 

.65 

0.99864  0.153 

0.99877  0.139 

0.99888  0.126 

0.99899  O.II4 

0.99909  0.103 

•7 

0.99823  0.139 

0.99840  0.126 

0.99855  0.114 

0.99869  0.103 

0.99882  0.094 

•75 

0.99788  O.I22 

0.99807  o.iio 

0.99826  o.ioo 

0.99843  0.090 

0.99858  0.082 

.8 

0.99757  o.ioi 

0.99780  0.092 

0.99801  0.083 

0.99820  O.O75 

0.99837  0.068 

.85 

0.99732  O.O78 

0.99758  0.071 

0.99781  0.064 

0.99802  O.O58 

0.99821  0.052 

•9 

0.99714  0.053 

0.99741  0.048 

0.99766  0.044 

0.99788  O.O39 

0.99809  0.036 

•95 

0.99703  0.027 

0.99732  0.024 

0-99757  0-022 

0.99780  0.020 

0.99801  0.018 

2.0 

0.99700  o.oo 

0.99728  o.oo 

0.99754  o.oo 

0.99777  o.oo 

0.99799  o.oo 

Note.     Negative  quantities  are  in  heavy  type. 

Examples,  tanh  (3.4  +  *  0.7)  = 

1.00131  /o°.io3  = 

1.00131  /o0.o6'.ii*. 

TABLE  XII.     HYPERBOLIC  TANGENTS,     tanh  (x  +  iq)  =  r  /r     CONTINUED 


=  3-5 


x  =  3-55 


*  =  3-6 


x  =  3-6S 


=  3-7 


o 

0.99818  o.oo 

0.99835  o.oo 

0.99851  o.oo 

0.99865  o.oo 

0.99878  o.oo 

0.05 

0.99820  0.016 

0.99837  0.015 

0.99853  0.013 

0.99867  O.OI2 

0.99879  o.on 

O.I 

0.99827  0.032 

0.99843  0.029 

0.99858  0.026 

0.99872  0.024 

0.99884  O.O22 

0.15 

0.99838  0.047 

0.99853  0.043 

0.99867  0.039 

0.99880  0.035 

0.99891  0.032 

O.2 

0.99853  0.06  1 

0.99867  0.055 

0.99879  0.050 

0.99891  0.046 

0.99901  0.041 

0.25 

0.99871  0.074 

0.99883  0.067 

0.98894  0.061 

0.99905  0.055 

0.99914  0.050 

0-3 

0.99893  0.084 

0.99903  0.076 

0.99912  0.069 

0.99921  0.063 

0.99928  0.057 

0-35 

0.99917  0.093 

0.99925  0.084 

0.99932  0.076 

0.99939  0.069 

0.99945  0.063 

0.4 

0.99944  0.099 

0.99949  0.090 

0.99954  0.081 

0.99958  0.074 

0.99962  0.067 

0-45 

0.99972  0.103 

0.99974  0.093 

0.99977  0.085 

0.99979  0.076 

0.99981  0.069 

0-S 

i.ooooo  0.104 

i.ooooo  0.095 

i.ooooo  0.086 

i.ooooo  0.077 

i.ooooo  0.070 

0-55 

1.00028  0.103 

1.00026  0.093 

1.00023  0.085 

I.OOO2I  0.076 

1.00019  0.069 

0.6 

1.00056  0.099 

1.00051  0.090 

1.00046  0.081 

1.00042  0.074 

1.00038  0.067 

0.65 

1.00083  o'°93 

1.00075  0.084 

.00068  0.076 

1.  0006  1  0.069 

1.00056  0.063 

0.7 

1.00107  0.084 

1.00097  0.076 

.00088  0.069 

1.00079  0.063 

1.00072  0.057 

0-75 

1.00129  0.074 

1.00117  0.067 

.00106  0.061 

1.00096  0.055 

1.00087  0.050 

0.8 

1.00148  0.061 

1.00134  0.056 

.OOI  2  1  O.O5O 

1.00109  0.046 

1.00099  0.041 

0.85 

1.00163  0.048 

1.00147  0.043 

.00133  0.039 

I.OOI2O  O.035 

1.00109  0.032 

0.9 

1.00174  0.032 

1.00157  0.029 

.00142  0.026 

1.00129  0.024 

I.OOII7  O.O22 

o-9S 

1.00180  0.016 

1.00163  0.015 

1.00148  0.013 

I.OOI34  O.OI2 

I.OOI2I  O.OII 

I.O 

1.00183  o'°o 

1.00165  °'°o 

1.00149  o.oo 

1.00135  o.oo 

I.OOI22  O.OO 

1.05 

1.00180  0.016 

1.00163  0.015 

1.00148  0.013 

I.OOI34  O.OI2 

I.OOI2I  O.OII 

i.i 

1.00174  0.032 

1.00157  0.029 

1.00142  0.026 

1.00129  0.024 

I.OOII7  O.O22 

i-iS 

1.00163  0.048 

1.00147  0.043 

1.00133  °-°30 

I.OOI2O  O.O35 

I.OOI09  O.O32 

1.2 

1.00148  0.061 

1.00134  0.056 

I.OOI2I  O.O5O 

1.00109  0.046 

I.OOO99  O.O4I 

1-25 

1.00129  0.074 

1.00117  0.067 

1.00106  0.061 

1.00096  0.055 

I.OOO87  O.O5O 

i-3 

1.00107  0.084 

1.00097  0.076 

1.00088  0.069 

1.00079  0.063 

I.OOO72  O.O57 

i-35 

1.00083  0.093 

1.00075  0.084 

1.00068  0.076 

1.  0006  1  0.069 

1.00056  O.O63 

1.4 

1.00056  0.099 

1.00051  0.090 

1.00046  0.08  1 

1.00042  0.074 

1.00038  O.067 

I-  45 

1.00028  0.103 

1.00026  0.093 

1.00023  0.085 

I.OOO2I  0.076 

I.OOOI9  O.069 

i-5 

i.ooooo  0.104 

i.ooooo  0.095 

i.ooooo  0.086 

i.ooooo  0.077 

i.ooooo  0.070 

i-55 

0.99972  0.103 

0.99974  0.093 

0.99977  0.085 

0.99979  0.076 

0.99981  0.069 

1.6 

0.99944  0.099 

0.99949  0.090 

0.99954  0.081 

0.99958  0.074 

0.99962  0.067 

1-65 

0.99917  0.093 

0.99925  0.084 

0.99932  0.076 

0.99939  0.069 

0.99945  0.063 

i-7 

0.99893  0.084 

0.99903  0.076 

0.99912  0.069 

0.99921  0.063 

0.99928  0.057 

i-7S 

0.99871  0.074 

0.99883  0.067 

0.99894  0.061 

0.99905  0.055 

0.99914  0.050 

1.8 

°-99853  0.061 

0.99867  0.055 

0.99879  0.050 

0.99891  0.046 

0.99901  0.041 

1.85 

0.99838  0.047 

0.99853  0.043 

0.99867  0.039 

0.99880  0.035 

0.99891  0.032 

1.9 

0.99827  0.032 

0.99843  0.029 

0.99858  0.026 

0.99872  0.024 

0.99884  O.O22 

i-9S 

0.99820  0.016 

0.99837  0.015 

0.99853  0.013 

0.99867  0.012 

0.99879  o.oii 

2.0        0.99818   o.oo        0.99835   o.oo        0.99851    o.oo        0.99865   o.oo        0.99878   o.oo 

Note.  Negative  quantities  are  in  heavy  type. 

Examples,     tanh  (3.6  -f  io)  =  0.99851  /o° 


tanh  (3.7  +  i  1.7)  =  0.99928  \o°.o57  =  0.99928  \o°.o3'.25". 


[136] 


TABLE  XII.     HYPERBOLIC  TANGENTS,     tanh  (x  +  iq)  =  r  /y.     CONTINUED 


x  =  . 

5-75 

x  = 

3-8 

x  =  ; 

[-85 

x  = 

3-9 

*  =  : 

•95 

? 

r 

7 

r 

7 

r 

7 

r 

7 

r 

7 

o 

o 

o 

0 

o 

0 

0.99889 

o.oo 

0.99900 

O.OO 

0.99909 

O.OO 

0.99918 

0.00 

0.99926 

O.OO 

0.05 

0.99891 

O.OIO 

0.99901 

0.009 

0.99911 

0.008 

0.99919 

0.007 

0.99927 

0.007 

O.I 

0.99895 

O.O2O 

0.99905 

0.018 

0.99914 

0.016 

0.99922 

0.015 

0-99930 

0.013 

0.15 

0.99902 

0.029 

0.99911 

0.026 

0.99919 

0.024 

0.99927 

O.O2I 

0-99934 

0.019 

0.2 

0.99911 

0.037 

0.99919 

0.034 

0.99927 

0.031 

0-99934 

O.O28 

0.99940 

0.025 

0.25 

0.99922 

0.045 

0.99929 

0.041 

0.99936 

0.037 

0.99942 

0.033 

0.99948 

0.030 

o-3 

0-99935 

0.051 

0.99941 

0.046 

0-99947 

0.042 

0.99952 

0.038 

0.99956 

0.034 

0.35 

0.99950 

0.056 

0-99955 

0.051 

0-99959 

0.046 

0.99963 

O.O42 

0.99966 

0.038 

0.4 

0.99966 

0.060 

0.99969 

0.055 

0.99972 

0.049 

0-99975 

0.045 

0.99977 

0.040 

0.45 

0.99983 

0.063 

0.99984 

0.057 

0.99986 

0.051 

0.99987 

0.046 

0.99988 

0.042 

o-5 

I.OOOOO 

0.063 

I.OOOOO 

0.057 

I.OOOOO 

0.052 

I.OOOOO 

0.047 

I.OOOOO 

0.043 

0-55 

1.00017 

0.063 

1.00016 

0.057 

1.00014 

0.051 

1.00013 

0.046 

I.OOOI2 

0.042 

0.6 

1.00034 

0.060 

1.00031 

0.055 

1.00028 

0.049 

1.00025 

0.045 

1.00023 

0.040 

0.65 

1.00050 

0.056 

1.00045 

0.051 

1.00041 

0.046 

1.00037 

O.042 

1.00034 

0.038 

0.7 

1.00065 

0.051 

1.00059 

0.046 

1.00053 

0.042 

1.00048 

0.038 

1.00044 

0.034 

o.7S 

1.00078 

0.045 

1.00071 

0.041 

1.00064 

0.037 

1.00058 

0.033 

1.00052 

0.030 

0.8 

1.00089 

0.037 

1.00081 

0.034 

1.00073 

0.031 

1.00066 

O.O28 

1.00060 

0.025 

0.85 

1.00099 

0.029 

1.00089 

0.026 

1.00081 

0.024 

1.00073 

O.O2I 

1.00066 

0.019 

0.9 

1.00105 

O.O2O 

1.00095 

0.018 

1.00086 

0.016 

1.00078 

O.OI5 

1.00071 

0.013 

0-95 

1.00109 

O.OIO 

1.00099 

0.009 

1.00089 

0.008 

1.00081 

O.OO7 

1.00073 

0.007 

I.O 

I.OOIII 

O.OO 

I.OOIOO 

O.OO 

1.00090 

O.OO 

.00082 

0.00 

1.00074 

0.00 

1.05 

1.00109 

O.OIO 

1.00099 

0.009 

1.00089 

0.008 

.00081 

0.007 

1.00073 

0.007 

i.i 

1.00105 

O.O2O 

1.00095 

0.018 

i.  '00086 

0.016 

.00078 

0.015 

1.00071 

0.013 

i.'S 

1.00099 

O.029 

1.00089 

0.026 

1.00081 

0.024 

.00073 

O.O2I 

1.00066 

0.019 

1.2 

1.00089 

0.037 

1.00081 

0.034 

1.00073 

0.031 

.00066 

O.O28 

1.00060 

0.025 

1-25 

1.00078 

0.045 

1.00071 

0.041 

1.00064 

0.037 

1.00058 

0-033 

.00052 

0.030 

1-3 

1.00065 

O.O5I 

1.00059 

0.046 

1.00053 

0.042 

1.00048 

0.038 

.00044 

0.034 

1-35 

1.00050 

O.O56 

1.00045 

0.051 

1.00041 

0.046 

1.00037 

O.042 

.00034 

0.038 

1.4 

1.00034 

O.06O 

1.00031 

0.055 

1.00028 

0.049 

1.00025 

0.045 

.00023 

0.040 

i  -45 

1.00017 

0.063 

1.00016 

0.057 

1.00014 

0.051 

1.00013 

0.046 

.00012 

0.042 

i-5 

I.OOOOO 

0.063 

I.OOOOO 

0.057 

I.OOOOO 

0.052 

I.OOOOO 

0.047 

I.OOOOO 

0.043 

i-SS 

0.99983 

O.O63 

0.99984 

0.057 

0.99986 

0.051 

0.99987 

0.046 

0.99988 

0.042 

1.6 

0.99966 

O.06O 

0.99969 

0.055 

0.99972 

0.049 

0-99975 

0.045 

0-99977 

0.040 

1.65 

0.99950 

0.056 

0-99955 

0.051 

0-99959 

0.046 

0.99963 

O.O42 

0.99966 

0.038 

i-7 

0-99935 

0.051 

0.99941 

0.646 

0-99947 

0.042 

0.99952 

O.O38 

0.99956 

0.034 

1-75 

0.99922 

0.045 

0.99929 

0.041 

0.99936 

0.037 

0.99942 

0.033 

0.99948 

0.030 

1.8 

0.99911 

0.037 

0.99919 

0.034 

0.99927 

0.031 

0-99934 

0.028 

0.99940 

0.025 

1.85 

0.99902 

O.O29 

0.99911 

0.026 

0.99919 

0.024 

0.99927 

O.O2I 

0-99934 

0.019 

1.9 

0.99895 

O.O2O 

0.99905 

0.018 

0.99914 

0.016 

0.99922 

O.OI5 

0.99930 

0.013 

i  -95 

0.99891 

O.OIO 

0.99901 

0.009 

0.99911 

0.008 

0.99919 

O.OO7 

0.99927 

0.007 

2.O 

0.99889 

O.OO 

0.99900 

0.00 

0.99909 

O.OO 

0.99918 

O.OO 

0.99926 

O.OO 

Note.  Negative  quantities  are  in  heavy  type. 

Examples,    tanh  (3.95  +  £0.9)  =  1.00071  /o°.oij 
tanh  (3.95  +  i  1.9) 


TABLE  XIII.     FUNCTIONS  OF  4  +  ig.    f  (4  +  »?)  =  «  + 


sinh 

cosh 

tanh 

9 

u      v 

U         V 

u      v 

3 

27.28992  o.oo 

27.30823  o.oo 

0.99933  o.oo 

3.05 

27.20579  2.14258 

27.22405  2.14114 

0.99934  o.oooio 

3.1 

26.95392  4-27195 

26.97202  4.26908 

0.99936  O.OOO2I 

3-IS 

26.53588  6.37498 

26.55370  6.37071 

0.99940  0.00030 

D.2 

25-95425  8.43871 

25.97166  8.43305 

0.99946  0.00039 

3.2S 

25.21260  10.45041 

25.22951  10.44340 

0.99953  0.00047 

3-3 

24.31551  12.39768 

24.33181  12.38935 

0.99961  0.00054 

5-35 

23.26848  14.26851 

23.28410  14.25895 

0.99970  0.00060 

3.4 

22.07800  16.05138 

22.09282  16.04061 

0.99979  0.00064 

5-45 

20.75141  17-73528 

20.76534  17.72339 

0.99989  0.00066 

3-5 

19.29688  19.30983 

19.30983  19.29688 

i.ooooo  0.00067 

3-55 

I7-72339  20.76534 

17.73528  20.75141 

i.ooon  0.00066 

3.6 

16.04061  22.09282 

16.05138  22.07800 

I.OOO2I  O.00064 

3.65 

14.25895  23.28410 

14.26851  23.26848 

i  .0003  o  o  .  00060 

3-7 

12.38935  24.33181 

12.39768  24.31551 

1.00039  0.00054 

3.75 

10.44340  25.22951 

10.45041  25.21260 

1.00047  0.00047 

3.8 

8.43305  25.97166 

8.43871  25.95425 

1.00054  0.00039 

3.85 

6.37071  26.55370 

6.37498  26.53588 

1.00060  0.00030 

3.9 

4.26908  26.97202 

4.27195  26.95392 

I.OOO64  O.OOO2I 

3-95 

2.14114  27.22405 

2.14258  27.20579 

1.00066  o.ooon 

I.O 

o.oo    27.30823 

o.oo    27.28992 

1.00067  o.oo 

•05 

2.14114  27.22405 

2.14258  27.20579 

1.00066  o.ooon 

.1 

4.26908  26.97202 

4.27195  26.95392 

I.OOO64  O.0002I 

•15 

6.37071  26.55370 

6.37498  26.53588 

1.00060  0.00030 

.2 

8.43305  25.97166 

8.43871  25.95425 

1.00054  0.00039 

•25 

10.44340  25.22951 

10.45041  25.21260 

1.00047  0.00047 

•3 

12.38935  24.33181 

12.39768  24.31551 

1.00039  0.00054 

•35 

14.25805  23.28410 

14.26851  23.26848 

1.00030  0.00060 

•4 

16.04061  22.09282 

16.05138  22.07800 

I.OOO2I  O.OOO64 

•45 

17.72339  20.76534 

17.73528  20.75141 

i.ooon  0.00066 

•5 

19.29688  19.30983 

19.30983  19.29688 

i.ooooo  0.00067 

•55 

20.75141  17.73528 

20.76534  17-72339 

0.99989  0.00066 

.6 

22.07800  16.05138 

22.09282  16.04061 

0.99979  0.00064 

•65 

23.26848  14.26851 

23.28410  14.25895 

0.99970  0.00060 

•7 

24.31551  12.39768 

24.33181  12.38935 

0.99961  0.00054 

•75 

25.21260  10.45041 

25.22951  10.44340 

0-99953  0.00047 

.8 

25-95425  8.43871 

25.97166  8.43305 

0.99946  0.00039 

.85 

26.53588  6.37498 

26.55370  6.37071 

0.99940  0.00030 

•9 

26.95392  4-27195 

26.97202  4.26908 

0.99936  O.O002I 

•95 

27.20579  2.14258 

27.22405  2.14114 

0.99934  o.oooio 

2.0  27.28992       0.00 


27.30823     o.oo 


0-99933    o.oo 


Note. 
Examples. 


Negative  quantities  are  in  heavy  type. 


sinh  (4  +  i  0.7)  = 
cosh  (4  +  *'  i. 25) 


12.38935  +  *  24.33181. 
*  —10.45041  +  i  25.21260. 


[138] 


TABLE  XIII.     FUNCTIONS  OF  4  +  iq.    /  (4  +  iq)  =  r 


sin' 

l 

cos 

h 

tan) 

1 

9 

f 

7 

r 

7 

r 

T 

o 

o 

o 

o 

27.28992 

o.oo 

27.30823 

o.oo 

0-99933 

o.oo 

0.05 

27.29002 

4-503 

27.30810 

4-497 

0.99934 

0.006 

O.I 

27.29036 

9.006 

27.30780 

8.994 

0.99936 

O.OI2 

0.15 

27.29090 

13-509 

27.30723 

I3-49I 

0.99940 

O.OlS 

O.2 

27.29166 

18.011 

27.30650 

17.989 

0.99946 

0.023 

0.25 

27.29260 

22.514 

27-3055° 

22.486 

0-99953 

O.027 

°-3 

27.29370 

27.016 

27.30445 

26.984 

0.99961 

0.031 

0-35 

27.29492 

31-517 

27.30324 

31-483 

0.99970 

0.034 

0.4 

27.29624 

36.018 

27.30190 

35-982 

0.99979 

0.037 

0-4S 

27.29764 

40.519 

27.30050 

40.481 

0.99990 

0.038 

o-5 

27.29908 

45.019 

27.29908 

44.981 

I.OOOOO 

0.038 

0-55 

27.30050 

49-519 

27.29764 

49.481 

I.OOOIO 

0.038 

0.6 

27.30190 

54.018 

27.29624 

53-982 

I.0002I 

0-037 

0.65 

27.30324 

58.517 

27.29492 

58-483 

1.00030 

0.034 

0.7 

27.30445 

63.016 

27.29370 

62.984 

1.00039 

0.031 

o.7S 

27-30550 

67.514 

27.29260 

67.486 

1.00047 

O.O27 

0.8 

27.30650 

72.011 

27.29166 

71.989 

1.00054 

0.023 

0.85 

27.30723 

76.509 

27.29090 

76.491 

1.00060 

O.OlS 

0.9 

27.30780 

81.006 

27.29036 

80.994 

1.00064 

O.OI2 

o.QS 

27.30810 

85-503 

27.29002 

85-497 

1.00066 

O.OO6 

1.0 

27.30823 

90 

27.28992 

90 

1.00067 

0.00 

1.05 

27.30810 

94-497 

27.29002 

94-503 

1.00066 

0.006 

i.i 

27.30780 

98.994 

27.29036 

99.006 

1.00064 

0.013 

i-i5 

27.30723 

103.491 

27.29090 

103.509 

1.00060 

O.OlS 

1.2 

27.30650 

107.989 

27.29166 

108.011 

1.00054 

O.O23 

I.2S 

27-30550 

112.486 

27.29260 

112.514 

1.00047 

O.027 

i-3 

27-30445 

116.984 

27.29370 

117.016 

1.00039 

0.031 

I-3S 

27.30324 

121.483 

27.29492 

121.517 

1.00030 

0.034 

1.4 

27.30190 

125.982 

27.29624 

126.018 

I.O002I 

0.037 

1-45 

27-30050 

130.481 

27.29764 

130-519 

I.OOOIO 

0.038 

•5 

27.29908 

134.981 

27.29908 

I35-019 

I.OOOOO 

0.038 

•55 

27.29764 

139.481 

27.30050 

139-519 

0.99990 

0.038 

.6 

27.29624 

143.982 

27.30190 

144.018 

0.99979 

0.037 

.65 

27.29492 

148.483 

27.30324 

148.517 

0.99970 

0.034 

•7 

27.29370 

152.984 

27.30445 

153.016 

0.99961 

O.O3I 

•75 

27.29260 

157.486 

27-30550 

I57-5I4 

0-99953 

O.O27 

.8 

27.29166 

161.989 

27.30650 

162.011 

0.99946 

O.O23 

1.85 

27.29000 

166.491 

27.30723 

166.509 

0.99940 

O.OlS 

1.9 

27.29036 

170.994 

27.30780 

171.006 

0.99936 

O.OI2 

1-95 

27.29002 

175-497 

27.30810 

I75-503 

0-99934 

O.006 

2.0        27.28992    180 


27.30823    180 


0-99933   °-°° 


Note.  Negative  quantities  are  in  heavy  type. 

Examples,     sinh  (4  +  *  i.o)  =  27.30823  /go0. 


tanh  (4  +  *  1.5)  =  i.oooo  \o°.038  =  i.oooo  \o°.o2'.i7* 


T'ABLE  XIV.     SEMI-EXPONENTIALS,          and  logw 


e*    .    e* 
x    -    log™  - 

e*    .    ex 
x    -    log.o  - 

e*    .    e* 

x     -    loglo  - 

4.00  27.299  1.4361479 
4.01  27.573  1.4404909 
4.02  27.851  1.4448338 
4.03  28.130  1.4491768 
4.04  28.413  1.4S3SI97 

4.50  45-009  1.6532952 
4.51  45.461  1.6576381 
4.52  45.918  1.6619811 
4-53  46.379  1.6663240 
4.54  46.845  1.6706669 

5.00  74-207  1.8704424 
5.01  74.952  1.8747854 
5.02  75.706  1.8791283 
5.03  76.467  1.8834712 
5.04  77.235  1.8878142 

4.05  28.699  1.4578627 
4.06  28.987  1.4622056 
4.07  29.278  1.4665485 
4.08  29.573  1.4708915 
4.09  29.870  1.4752344 

4-55  47-3  1(5  1.6750099 
4.56  47-792  1.6793528 
4.57  48.272  1.6836958 
4.58  48.757  1.6880387 
4.59  49.247  1.6923817 

5.05  78.011  1.8921571 
5.06  78.795  1.8965001 
5-07  79-587  1.9008430 
5.08  80.387  1.9051860 
5.09  81.195  1.9095289 

4.10  30.170  1.4795774 
4.11  30.473  1.4839203 
4.12  30.780  1.4882633 
4.13  31.089  1.4926062 
4.14  31.401  1.4969492 

4.60  49.742  1.6967246 
4.61  50.242  1.7010676 
4.62  50.747  1.7054105 
4.63  51.257  1.7097535 
4.64  51.772  1.7140964 

5.10  82.011  1.9138719 
5.11  82.835  1.9182148 
5.12  83.668  1.9225577 
5.13  84.509  1.9269007 
5.14  85.358  1.9312436 

4.15  31.717  1.5012921 
4.16  32.036  1.5056350 
4.17  32.358  1.5099780 
4.18  32.683  1.5143209 
4.19  33.011  1.5186639 

4.65  52.292  1.7184393 
4.66  52.818  1.7227823 
4-67  53-349  1.7271252 
4.68  53.885  1.7314682 
4.69  54-427  1.7358m 

5.15  86.216  1.9355866 
5.16  87.082  1.9399295 
5.17  87.957  1.9442725 
5.18  88.841  1.9486154 
5.19  89.734  1.9529584 

4-20  33.343  1-5230068 

4.21  33.678  1.5273498 
4.22  34.017  1.5316927 

4-23  34-359  1-5360357 
4.24  34.704  1.5403786 

4.70  54-974  I-740IS4I 
4.71  55-526  1.7444970 
4.72  56.084  1.7488400 
4.73  56.648  1.7531829 
4.74  57.217  1.7575258 

5.20  90.636  1.9573013 
5.21  91.547  1.9616443 
5.22  92.467  1.9659872 
5-23  93.396  1.9703301 
5.24  94.335  I-974673I 

4-25  35-053  I.54472I5 
4.26  35.405  1.5490645 

4-27  35.76i  1.5534074 
4.28  36.120  1.5577504 
4.29  36.483  1.5620933 

4-75  57-792  1.7618688 
4-76  58.373  1.7662117 
4.77  58.960  1.7705547 
4.78  59.552  1.7748976 
4.79  60.151  1.7792406 

5.25  95.283  1.9790160 
5.26  96.241  1.9833590 
5.27  97.208  1.9877019 
5.28  98.185  1.9920449 
5.29  99.172  1.9963878 

4.30  36.850  1.5664363 
4.31  37.220  1.5707792 
4-32  37-594  1.5751222 
4-33  37-972  1.5794651 
4-34  38.354  1.5838081 

4.80  60.755  1-7835835 
4.81  61.366  1.7879265 
4.82  61.983  1.7922694 
4.83  62.605  1.7966123 
4.84  63.235  1.8009553 

5.30  100.168  2.0007308 
5.31  101.175  2.0050737 
5.32'  102.192  2.0094166 
5.33  103.219  2.0137596 
5.34  104.256  2.0181025 

4-35  38.739  1.588151° 
4.36  39.129  1.5924939 
4-37  39-522  1.5968369 
4-38  39-919  1.6011798 
4-39  40-320  1.6055228 

4.85  63.870  1.8052982 
4.86  64.512  1.8096412 
4.87  65.160  1.8139841 
4.88  65.815  1.8183271 
4.89  66.477  1-8226700 

5-35  105.304  2.0224455 
5.36  106.362  2.0267884 
5.37  107.431  2.0311314 
5.38  108.511  2.0354743 
5.39  109.602  2.0398173 

4.40  40.725  1.6098657 
4.41  41.135  1.6142087 
4.42  41-548  1.6185516 
4.43  41.966  1.6228946 
4-44  42.387  1.6272375 

4.90  67.145  1.8270130 
4.91  67.820  1.8313559 
4.92  68.501  1.8356989 
4.93  69.190  1.8400418 
4.94  69.885  1.8443847 

5.40  110.703  2.0441602 
5.41  111.816  2.0485031 
5.42  112.940  2.0528461 
5.43  114-075  2.0571890 
5.44  115.221  2.0615320 

4.45  42.813  1.6315804 
4.46  43.244  1.6359234 
4.47  43-678  1.6402663 
4.48  44.117  1.6446093 
4.49  44-561  1.6489522 

4.95  70.587  1.8487277 
4.96  71.297  1.8530706 
4.97  72.013  1.8574136 
4.98  72.737  1.8617565 
4.99  73.468  1.8660995 

5.45  116.379  2.0658749 
5.46  117-549  2.0702179 
5.47  118.730  2.0745608 
5.48  119.923  2.0789038 
5.49  121.129  2.0832467 

4.50  45-009  1.6532952 

5.00  74.207  1.8704424 

5.50  122.346  2.0875897 

e4'*0  /c4'20\ 

Example.      —  =  33.343        logio  \—)  =  1.5230068. 

[140] 


TABLE  XIV.     SEMI-EXPONENTIALS.     -  and  loglo  (  -  ).     CONTINUED 


*   7   log">  7 

ex    .    e* 
x         Iog10  - 

2           2 

e*    .    ex 
x    -    loglo  - 

5.50  122.346  2.0875897 
5.51  123.576  2.0919326 
5.52  124.818  2.0962755 
5.53  126.072  2.1006185 

5-54  127.339  2.1049614 

6.00  201.714  2.3047369 
6.01  203.742  2.3090785 
6.02  205.789  2.3134222 
6.03  207.858  2.3177667 
6.04  209.947  2.3221098 

6-5°  332-57I  2.5218844 
6-51  335.9I3  2.5262268 
6.52  339-289  2.5305699 
6.53  342.699  2.5349128 
6.54  346.143  2.5392556 

5.55  128.619  2.1093044 
5.56  129.911  2.1136473 
5.57  131.217  2.1179903 
5.58  132.536  2.1223332 
5.59  133.868  2.1266762 

6.05  212.057  2.3264527 
6.06  214.188  2.3307951 
6.07  216.340  2.3351368 
6.08  218.514  2.3394793 
6.09  220.711  2.3438220 

6-55  349-622  2.5435988 
6-56  353.135  2.5479408 
6-57  356-685  2.5522849 
6.58  360.270  2.5566281 
6-59  363-890  2.5609701 

5.60  135-213  2.1310191 
5.61  136.572  2.1353620 
5.62  137-945  2.1397050 
5.63  139.331  2.1440479 
5.64  140.731  2.1483909 

6.10  222.929  2.3481666 
6.  it  225.169  2.3525086 
6.12  227.432  2.3568516 
6.13  229.718  2.3611951 
6.14  232.027  2.3655386 

6.60  367.547  2.5653129 
6.61  371.241  2.5696560 
6.62  374.973  2.5740000 
6.63  378.741  2.5783424 
6.64  382.547  2.5826849 

5.65  142.146  2.1527338 
5.66  143-574  2.1570768 
5.67  i45-OI7  2.1614197 
5.68  146.475  2.1657627 
5.69  147.947  2.1701056 

6.15  234.359  2.3698817 
6.16  236.714  2.3742240 
6.17  239.093  2.3785669 
6.18  241.496  2.3829100 
6.19  243.923  2.3872527 

6.65  386.391  2.5870270 
6.66  390.275  2.5913708 
6.67  394.197  2.5957134 
6.68  398.160  2.6000576 
6.69  402.161  2.6044000 

5.70  149-434  2.1744485 
5.71  150.936  2.1787915 
5.72  152.452  2.1831344 
5-73  I53-985  2.1874774 
5-74  I55-532  2.1918203 

6.20  246.375  2.3915966 
6.21  248.851  2.3959394 

6.22  251.352  2.4OO2824 
6.23  253.877  2.4046234 
6.24  256.429  2.4089672 

6.70  406.202  2.6087420 
6.71  410.285  2.6130856 
6.72  414.409  2.6174292 
6.73  418.574  2.6217721 
6.74  422.780  2.6261144 

5.75  157-095  2.1961633 
5.76  158.674  2.2005062 
5.77  160.269  2.2048492 
5.78  161.880  2.2091921 
5-79  163.507  2.2135351 

6.25  259.006  2.4133099 
6.26  261.609  2.4176526 
6.27  264.239  2.4219970 
6.28  266.894  2.4263388 
6.2Q  269.576  2.4306813 

6.75  427.030  2.6304584 
6.76  431.321  2.6348006 
6.77  435-656  2.6391436 
6.78  440.034  2.6434862 
6.79  444.457  2.6478298 

5.80  165.150  2.2178780 
5.81  166.810  2.2222209 
5.82  168.486  2.2265639 
5.83  170.179  2.2309068 
5.84  171.890  2.2352498 

6.30  272.285  2.4350237 
6.31  275.022  2.4393675 
6.32  277.786  2.4437104 
6.33  280.578  2.4480536 
6.34  283.398  2.4523967 

6.80  448.923  2.6521719 
6.81  453-435  2.6565151 
6.82  457-993  .2.6608589 
6.83  462.595  2.6652009 
6.84  467-244  2.6995437 

5.85  173-617  2.2395927 
5.86  175-362  2.2439357 
5.87  177.124  2.2482786 
5.88  178.905  2.2526216 
5.89  180.703  2.2569645 

6.35  286.246  2.4567394 
6.36  289.123  2.4610826 
6.37  292.029  2.4654260 
6.38  294.964  2.4697690 
6.39  297.928  2.474III4 

6.85  471.940  2.6738868 
6.86  476.683  2.6782296 
6.87  481.474  2.6825728 
6.88  486.312  2.6869150 
6.89  491.200  2.6912584 

5.90  182.519  2.2613074 
5.91  184.353  2.2656504 
5.92  186.206  2.2699933 
5.93  188.077  2.2743363 
5.94  189.967  2.2786792 

6.40  300.922  2.4784540 
6.41  303.947  2.4827979 
6.42  3O7.OO2  2.4871412 
6.43  310.087  2.4914836 
6.44  313.203  2.4958260 

6.90  496.137  2.6956016 
6.91  501.123  2.6999443 
6.92  506.160  2.7042878 
6.93  511.246  2.7086299 
6.94  516.386  2.7129744 

5.95  191.877  2.2830222 
5.96  193.805  2.2873651 
5.97  195.753  2.2917081 
5.98  197.720  2.2960510 
5.99  199.707  2.3003939 

6.45  316.352  2.5001705 
6.46  319.530  2.5045II6 
6.47  322.742  2.5088555 
6.48  325.985  2.5I3I977 
6.49  329.262  2.5I754I6 

6.95  521.575  2.7173168 
6.96  526.816  2.7216589 
6-97  532.112  2.7260030 
6.98  537-459  2.7303454 
6.99  542.860  2.7346878 

6.00  201.714  2.3047369 

6-50  332.571   2.5218844 

7.00  548.317  2.7390317 

gb.60 

Example.          -  =  135-213 

2 


>- 


1310191. 


[141] 


FABLE  XIV.  SEMI-EXPONENTIALS.   ^  and  Iogi0  (-\  CONTINUE 

ez        ex 
x         log  10  — 

2             2 

ex        e* 
x          log  10  — 

2             2 

e'        e' 
x     -     loglo  - 

7.00  548.317  2.7390317 
7.01  553.827  2.7433741 

7.02  559-393  2.7477170 
7.03  565.015  2.7520600 
7.04  570.694  2.7564033 

7.50   904.021   2.9561785 
7.51    913.107  2.9605217 
7.52   922.284  2.9648647 

7-53  931-553  2.9692076 
7.54  940.915  2.9735504 

8.00  1490.479  3.1733259 
8.01  1505.457  3.1776683 
8.02  1520.589  3.1820118 
8-03  1535.870  3.1863546 
8.04  1551.306  3.1906976 

7.05  576.429  2.7607458 
7.06  582.223  2.7650893 
7.07  588.074  2.7694320 
7.08  593-984  2.7737747 
7.09  599.954  2.7781180 

7-55  950.371  2.9778932 
7.56  959.923  2.9822364 
7-57  969-570  2.9865792 
7.58  979-3I4  2.9909220 
7.59  989.157  2.9952653 

8.05  1566.895  3-1950399 
8.06  1582.645  3.1993836 
8.07  1598.552  3.2037268 
8.08  1614.617  3.2080694 
8.09  1630.841  3.2124116 

7.10  605.984  2.7824612 
7.11  612.074  2.7868039 
7.12  618.225  2.7911467 
7.13  624.439  2.7954901 
7.14  630.714  2.7998325 

7.60  999.098  2.9996080 
7.61  1009.139  3.0039510 
7.62  1019.281  3.0082939 
7.63  1029.525  3.0126369 
7.64  1039.872  3.0169799 

8.10  1647.234  3.2167554 
8.ii  1663.789  3.2210981 
8.12  1680.510  3.2254412 
8.13  1697.400  3.2297842 
8.14  1714.458  3.2341270 

7.15  637.053  2.8041755 
7.16  643.456  2.8085189 
7.17  649.922  2.8128612 
7.18  656.454  2.8172043 
7.19  663.052  2.8215476 

7.65  1050.323  3.0213229 
7.66  1060.879  3.0256659 
7.67  1071.541  3.0300088 
7.58  1082.310  3.0343517 
7.69  1093.187  3.0386944 

8.15  1731-690  3.2384703 
8.16  1749.092  3.2428130 
8.17  1766.672  3.2471560 
8.18  1784.427  3.2514988 
8.19  1802.364  3-2558425 

7.20  669.715  2.8258901 
7.21  676.446  2.8302331 
7.22  683.245  2.8345765 
7.23  690.111  2.8389189 
7.24  697.047  2.8432620 

7.70  1104.174  3.0430376 
7.71  1115.271  3.0473806 
7.72  1126.480  3.0517234 
7-73  ii37-8oi  3.0560663 
7.74  1149.236  3.0604092 

8.20  1820.476  3.2601848 
8.21  1838.774  3.2645284 

8.22  1857.251  3.2688706 
8.23  1875.914  3.2732129 
8.24  1894.770  3-2775566 

7.25  704.052  2.8476047 
7.26  711.128  2.8519478 
7.27  718.275  2.8562908 
7.28  725-494  2.8606338 
7.29  732.785  2.8649766 

7.75  1160.786  3.0647523 
7.76  1172.452  3.0690950 
7.77  1184.236  3.0734383 
7.78  1196.137  3.0777810 
7.79  1208.159  3.0821242 

8.25  1913.812  3.2188994 
8.26  1933.047  3.2862424 
8.27  1952.473  3.2905850 
8.28  1972.098  3.2949284 
8.29  1991.913  3.2992704 

7.30  740.150  2.8693197 
7.31  747.589  2.8^36629 
7.32  755.102  2.8780056 
7.33  762.691  2.8823487 
7-34  770.356  2.8866915 

7.80  1220.301  3.0864670 
7.81  1232.565  3.0908098 
7.82  1244.953  3.0951531 
7.83  1257.465  3.0994961 
7.84  1270.102  3.1038386 

8.30  2011.936  3.3036142 
8.31  2032.158  3-3079575 
8.32  2052.580  3.3123000 
8.33  2073.206  3.3166425 
8.34  2094.045  3.3209860 

7-35  778-098  2.8910343 
7.36  785.918  2.8953772 
7-37  793-8i7  2.8997205 
7.38  801.795  2.9040633 
7.39  809.853  2.9084062 

7.85  1282.867  3.1081818 
7.86  1295.760  3.1125246 
7.87  1308.783  3.1168677 
7.88  1321.936  3.1212105 
7.89  1335.222  3-1255535 

8.35  2115.092  3-3253293 
8.36  2136.347  3.3296718 
8.37  2157.819  3.3340150 
8.38  2179.505  3-3383578 
8.39  2201.409  3.3427008 

7.40  817.992  2.9127491 
7.41  826.213  2.9170920 
7.42  834.517  2.9214351 
7.43  842.904  2.9257782 
7.44  851.375  2.9301209 

7.90  1348.641  3.1298964 
7.91  1362.195  3.1342394 
7.92  1375-886  3.1385826 
7.93  1389.713  3.1429254 
7.94  1403.680  3.1472680 

8.40  2223.533  3-3470436 
8.41   2245.881  3-35I3868 
8-42  2268.452  3.3557296 
8.43  2291.250  3.3600725 

8-44  2314.277  3.3644154 

7.45  859.932  2.9344641 
7.46  868.574  2.9388068 
7-47  877.303  2.9431496 
7.48  886.  120  2.9474925 
7.49  895.026  2.9518356 

7.95  1417-787  3-1516110 
7.96  1432-036  3-1559539 
7.97  1446.429  3.1602971 
7.98  1460.966  3.1646402 
7.99  1475.648  3.1689827 

8-45  2337.536  3-3687583 
8.46  2361.030  3.3731014 
8.47  2384.752  3-3774433 
8.48  2408.725  3.3817872 
8.49  2432.926  3.3861290 

7.50  904.021  2.9561785 

8.00  1490-479  3-I733259 

8.50  2457.383  3-3904730 

e7-10 
Example. 

2 

/C7.1(K 

=  605.984    log  10  (^  , 

)  =  2.7824612. 

[142] 


TABLE  XIV.     SEMI-EXPONENTIALS.     -  and  Iog10  (  ^ 


CONTINUED 


X 

e* 

i    e* 
log  10  — 

X 

e* 

log  10  - 

X 

e? 

g* 

log  10  - 

2 

2 

2 

2 

2 

2 

8.50 

2457.383 

3.3904730 

9-00 

4051.543 

3.6076204 

9-50 

6679.863 

3.8247676 

8.51 

2482.082 

3.3948162 

9-OI 

4092.263 

3.6119636 

9-51 

6746.988 

3.8291101 

8.52 

2507.027 

3.3991590 

9-02 

4I33-388 

3.6163062 

9-52 

6814.805 

3-8334534 

8-53 

2532.221 

3.4035016 

9-°3 

4174.929 

3.620649! 

9-53 

6883.295 

3.8377964 

8-54 

2557.672 

3.4078448 

9.04 

4216.889 

3.6249922 

9-54 

6952475 

3.8421394 

8-55 

2583-380 

34I2I882 

9-°S 

4259.264 

3.6293345 

9-55 

7022.345 

3.8464822 

8.56 

2609.341 

3.4165308 

9.06 

4302.076 

3.6336780 

9-56 

7092.923 

3.8508252 

8-57 

2635.562 

3.4208732 

9.07 

4345.302 

3.6380200 

9-57 

7164.203 

3.8551679 

8.58 

2662.052 

3.4252166 

9.08 

4388.982 

3.6423638 

9-58 

7236.210 

3.8595112 

8-59 

2688.810 

3.4295601 

9.09 

4433-098 

3.6467073 

9-59 

7308.929 

3.8638537 

8.60 

2715.830 

3.4339026 

9.10 

4477.646 

3.6510498 

9.60 

7382.390 

3.8681970 

8.61 

2743.126 

3.4382458 

9.11 

4522.647 

3.6553927 

9.61 

7456.583 

3.8725398 

8.62 

2770.693 

3.4425884 

9.12 

4568.100 

3-6597356 

9.62 

753I-526 

3.8768830 

8.63 

2798.535 

3.4469308 

9-13 

4614.016 

3.6640791 

9-63 

7607.221 

3.8812260 

8.64 

2826.665 

3.4512744 

9.14 

4660.383 

3.6684216 

9.64 

7683.672 

3.8855688 

8.65 

2855.070 

3.4556167 

9-iS 

4707.211 

3.6727637 

9.65 

7760.882 

3.8899111 

8.66 

2883.767 

3.4599602 

9.16 

4754.528 

3.6771074 

9.66 

7838.890 

3.8942546 

8.67 

2912.745 

3.4643025 

9.17 

4802.308 

3.6814500 

9.67 

7917.680 

3.8985980 

8.68 

2942.023 

3.4686462 

9.18 

4850.577 

3.6857934 

9.68 

7997.247 

3.9029406 

8.69 

2971.592 

3.4729891 

9.19 

4899.328 

3.6901365 

9.69 

8077.622 

3.9072835 

8.70 

3001.456 

34773320 

9.20 

4948.563 

3.6944792 

9.70 

8158.802 

3.9116264 

8.71 

3031.621 

3.4816749 

9.21 

4998.284 

3.6988209 

9.71 

8240.792 

3.9159690 

8.72 

3062.088 

3.4860178 

9.22 

5048.532 

3.7031652 

9.72 

8323.623 

3.9203124 

8-73 

3092.852 

3.4903592 

9-23 

5099.272 

3.7075082 

9-73 

8407.262 

3.9246546 

8.74 

3123.948 

3.4947038 

9.24 

5150.519 

3.7II85IO 

9-74 

8491.770 

3.9289982 

8-75 

3*55-337 

3.4990458 

9-25 

5202.272 

3.7161930 

9-75 

8577.112 

3-93334II 

8.76 

3187.054 

3.5033896 

9.26 

5254.569 

3.7205370 

9.76 

8663.316 

3.9376842 

8-77 

3219.085 

3-5077325 

9.27 

5307-367 

3.7248791 

9-77 

8750.384 

3.9420270 

8.78 

3251.440 

3.5120756 

9.28 

5360.716 

3.7292228 

9.78 

8838.326 

3.9463700 

8.79 

3284.114 

3.5164182 

9.29 

54I4-587 

3-7335654 

9-79 

8927.154 

3.9507131 

8.80 

3317.122 

3.5207614 

9-30 

5469.009 

3.7379086 

9.80 

9016.875 

3.9550560 

8.81 

3350.460 

3.5251044 

9-31 

5523-975 

3.7422517 

9.81 

9107.481 

3.9593983 

8.82 

3384-I33 

3-5294474 

9-32 

5579-491 

3.7465946 

9.82 

9199.026 

3.9637418 

8.83 

3418.141 

3.5337900 

9-33 

5635-563 

3-7509373 

9-83 

9291.480 

3.9680850 

8.84 

3452.496 

3-538I332 

9-34 

5692.203 

3.7552804 

9.84 

9384.860 

3.9724278 

8.85 

3487.197 

3.5424766 

9-35 

5749405 

3.7596229 

9-85 

9479.163 

3.9767701 

8.86 

3522.243 

3.5468192 

5807.194 

3.7639664 

9.86 

9574-444 

3.9811136 

8.87 

3557-63I 

3.5511609 

9-37 

5865.555 

3.7683091 

9.87 

9670.678 

3.9854569 

8.88 

3593-395 

3-5555050 

9-38 

5924.507 

3.7726522 

9.88 

9767.860 

3.9897994 

8.89 

3629.512 

3.5598482 

9-39 

5984.054 

3.7769956 

9.89 

9866.020 

3.9941420 

8.90 

3665-986 

3.5641908 

9.40 

6044.19! 

3.7813382 

9.90 

9965.186 

3.9984854 

8.91 

3702.820 

3.5685326 

9.41 

6104.922 

3.7856801 

9.91 

10065.350 

4.0028289 

8.92 

3740.045 

3.5728768 

9.42 

6l66.29O 

3.7900240 

9.92 

10166.494 

4.007I7I2 

8-93 

3777-635 

3.5772201 

9-43 

6228.269 

3.7943674 

9-93 

10268.667 

4.0II5I4I 

8.94 

38I5-597 

3.5815626 

9-44 

6290.860 

3.7987100 

9-94 

10371.873 

4.0158572 

8-95 

3853-937 

3.5859044 

9-45 

6354.080 

3.8030526 

9-95 

10476.107 

4.0201999 

8.96 

3892.678 

3.5902486 

9.46 

64I7-943 

3.8073958 

9.96 

10581.397 

4.0245430 

8-97 

3931-795 

3.5945909 

9-47 

6482.450 

3.8II7392 

9-97 

10687.745 

4.0288860 

8.98 

3.5989344 

9.48 

6547.591 

3.8l6o8l6 

9.98 

10795.160 

4.0332290 

8.99 

4011.228 

3.6032773 

9-49 

6613.388 

3.8204240 

9-99 

10903.652 

4.0375721 

9.00 

4051.543 

3.6076204 

9-50 

6679.863 

3.8247676 

IO.OO 

11013.233 

4.0419148 

gS-M 

Example. 

i 
=  3665.986 

log  10  (  

)  =  3 

.5641908. 

[143] 

TABLE  XV 
REAL  HYPERBOLIC  FUNCTIONS.    /  (x  +  io)  =  u  +  to 


Sinh0 


Cosh0        Tanh0        Cothe       Sech0       Cosech  6      0 


o.oo 

o.oo 

I.OO 

o.oo 

00 

I.OO 

00 

0.00 

O.OI 

O.OIOOOO 

1.000050 

O.OIOOO 

IOO. 

0.9999 

IOO. 

O.OI 

0.02 

O.O20OOI 

I.OOO2OO 

O.O2OOO 

50. 

0.9998 

50. 

O.O2 

0.03 

0.030005 

I.OOO45O 

0.02999 

33-34 

0.9995 

33-333 

0.03 

0.04 

0.04001  1 

1.000800 

0.03998 

25.013 

0.9992 

24.99 

O.O4 

O.OS 

0.050021 

I.OOI25O 

0.04996 

20.016 

0.9987 

19.992 

0.05 

0.06 

0.060036 

1.001801 

0.05993 

16.686 

0.9982 

16.657 

O.O6 

O.O7 

0.070057 

1.002451 

0.06989 

14.308 

0-9975 

14.274 

O.O7 

0.08 

0.080085 

1.003202 

0.07983 

12.527 

0.9968 

12.487 

0.08 

O.OQ 

0.090122 

1.004053 

0.08976 

11.141 

0-9959 

11.097 

0.09 

O.IO 

0.100167 

1.005004 

0.09967 

10.033 

0.9950 

9-983 

O.IO 

O.II 

O.IIO222 

1.006056 

0.10956 

9.128 

0.9940 

9-073 

O.II 

O.I  2 

0.120288 

1.007209 

0.11943 

8.373 

0.9928 

8.314 

O.I  2 

0.13 

0.130366 

1.008462 

0.12927 

7-735 

0.9916 

7.669 

0.13 

0.14 

0.140458 

1.009816 

0.13909 

7.189 

0.9902 

7.120 

O.I4 

O.IS 

0.150563 

1.011271 

0.14888 

6.716 

0.9888 

6.642 

0.15 

0.16 

0.160684 

1.012827 

0.15865 

6.303 

0.9873 

6.223 

0.16 

0.17 

O.I7O82O 

1.014485 

0.16838 

5-939 

0.9857 

5-854 

0.17 

0.18 

0.180974 

1.016244 

0.17808 

5-6i5 

0.9840 

5-525 

0.18 

0.19 

O.I9II45 

1.018104 

0.18775 

5-325 

0.9822 

S-232 

0.19 

O.2O 

0.201336 

1.020067 

0.19737 

5.067 

0.9803 

4.967 

O.2O 

O.2I 

0.2II547 

1.022131 

0.20696 

4.832 

0.9784 

4.726 

O.2I 

O.22 

O.22I779 

1.024298 

0.21652 

4.618 

0.9763 

4-509 

O.22 

0.23 

0.232033 

1.026567 

0.22603 

4-425 

0.9742 

4-3io 

0.23 

O.24 

O.2423II 

1.028939 

0.23549 

4.246 

0.9719 

4.127 

O.24 

0.25 

O.2526I2 

1.031413 

0.24492 

4.083 

0.9695 

3-959 

0.25 

O.26 

0.262939 

1.033991 

0.25430 

3-932 

0.9671 

3-803 

0.26 

O.27 

0.273292 

1.036672 

0.26363 

3-793 

0.9646 

3-659 

O.27 

0.28 

0.283673 

1.039457 

0.27290 

3.664 

0.9620 

3-52S 

0.28 

0.29 

0.294082 

1.042346 

0.28214 

3-544 

0.9591 

3.400 

O.29 

0.30 

0.304520 

1-045339 

0.29131 

3-433 

0.9566 

3-284 

0.30 

0.31 

0.314989 

1.048436 

0.30043 

3-328 

0-9537 

3-r75 

0.31 

0.32 

0.325489 

1.051638 

0.30951 

3-231 

0.9511 

3-072 

0.32 

0-33 

0.336022 

1.054946 

0.31852 

3.140 

0.9479 

2.976 

0-33 

0-34 

0.346589 

1-058359 

0.32748 

3-053 

Q-9447 

2.885 

o-34 

0-35 

0.357190 

1.061878 

0.33637 

2-973 

0.9416 

2.800 

o-35 

0.36 

0.367827 

1.065503 

0.34522 

2.897 

0-9385 

2,719 

0.36 

0-37 

0.378500 

1.069234 

0.35399 

2.825 

0-9353 

2.642 

0-37 

0.38 

0.389212 

1-073073 

0.36271 

2-757 

0.9319 

2.569 

0.38 

0-39 

0.399962 

1.077019 

0.37136 

2.693 

0.9285 

2.500 

o-39 

0.40 

0.410752 

1.081072 

0.37995 

2.632 

0.9250 

2-434 

0.40 

O.4I 

0.421584 

1.085234 

0.38847 

2-574 

0.9215 

2.372 

0.41 

O.42 

0-432457 

1.089504 

0.39693 

2-519 

0.9178 

2.312 

0.42 

0-43 

0-443374 

1.093883 

0.40532 

2.467 

0.9141 

2.256 

0-43 

0.44 

0-454335 

1.098372 

0.41365 

2.417 

0.9103 

2.2OI 

0.44 

0-45 

0.465342 

1.102970 

0.42190 

2-370 

0.9066 

2.149 

0-45 

0.46 

0.476395 

1.107679 

0.43009 

2-325 

0.0925 

2.099 

0.46 

0.47 

0.487496 

1.112498 

0.43820 

2.282 

0.8988 

2.051 

0.47 

0.48 

0.498646 

1.117429 

0.44624 

2.241 

0.8949 

2.O06 

0.48 

0.49 

0.509845 

1.122471 

0.45421 

2.2O2 

0.8909 

I.96l 

0-49 

Example.      sinh  0.25  =  0.252612. 


TABLE  XV 
REAL  HYPERBOLIC  FUNCTIONS.    /  (x  +  io)  =  u  +  io.     CONTINUED 


e 

Sinhfl 

Coshtf 

Tanh0 

Cothfl 

Sechfl 

Cosech  & 

e 

0.50 

0.521095 

1.127626 

0.46211 

2.164 

0.8868 

1.919 

0.50 

0.51 

0-532398 

1.132893 

0.46995 

2.128 

0.8827 

1.878 

0.51 

0.52 

0-543754 

1.138274 

0.47769 

2.093 

0.8785 

1.839 

0.52 

o-S3 

o-555i64 

1.143769 

0.48538 

2.060 

0.8743 

1.801 

o-53 

0-54 

0.566629 

1.149378 

0.49299 

2.028 

0.8700 

1-765 

0-54 

0-55 

0.578152 

1.155101 

0.50052 

1.998 

0.8658 

1-730 

0-55 

0.56 

0-589732 

1.160941 

0.50797 

1.969 

0.8614 

1.696 

0.56 

0-57 

0.601371 

1.166896 

0-51536 

1.940 

0.8570 

1.663 

0-57 

0.58 

0.613070 

1.172968 

0.52266 

i-9i3 

0.8525 

1.631 

0.58 

o-59 

0.624831 

1.179158 

0.52990 

1.887 

0.8480 

1.601 

0-59 

0.60 

0.636654 

1.185465 

0.53704 

1.862 

0.8435 

I-57I 

0.60 

0.61 

0.648540 

1.191891 

0.54413 

1-838 

0.8390 

1.542 

0.61 

0.62 

0.660492 

1.198436 

0.55112 

1.814 

0.8344 

i.SH 

0.62 

0.63 

0.672509 

1.205101 

0.55805 

1.792 

0.8298 

1.487 

0.63 

0.64 

0.684504 

1.211887 

0.56490 

1.770 

0.8251 

1.461 

0.64 

0.65 

0.696748 

1.218793 

0.57166 

1.749 

0.8205 

1-435 

0.65 

6.66 

0.708970 

1.225822 

0.57836 

1.729 

0.8158 

1.410 

0.66 

0.67 

0.721264 

1.232973 

0.58498 

1.709 

0.8110 

1-387 

0.67 

0.68 

0.733630 

1.240247 

0.59152 

1.690 

0.8065 

1-363 

0.68 

0.69 

0.746070 

1.247646 

0.59798 

1.672 

0.8015 

1.340 

0.69 

0.70 

0.758584 

1.255169 

0.60437 

1-655 

0.7967 

1.318 

0.70 

0.71 

0.771174 

1.262818 

0.61067 

1.637 

0.7919 

1.297 

0.71 

0.72 

0.783840 

1.270593 

0.61691 

1.621 

0.7870 

1.276 

0.72 

o.73 

0.796586 

1.278495 

0.62306 

1.605 

0.7821 

1.255 

0-73 

0.74 

0.809411 

1.286525 

0.62914 

1.590 

0-7773 

1.235 

0.74 

o.7S 

0.822317 

1.294683 

0.63516 

1-5744 

0.7724 

i.  216 

0-75 

0.76 

0.835305 

1.302971 

0.64108 

1-5599 

0.7675 

1.1972 

0.76 

0.77 

0.848377 

1.311390 

0.64693 

1-5457 

0.7625 

1.1787 

0.77 

0.78 

0.861533 

1.319939 

0.65271 

1-5320 

0.7576 

1.1607 

0.78 

0.79 

0.874776 

1.328621 

0.65842 

1.5188 

0.7527 

1.1431 

0.79 

0.80 

0.888106 

1-337435 

0.66403 

I.5059 

0.7477 

1.1259 

0.80 

0.81 

0.901525 

1-346383 

0.66959 

1-4934 

0.7427 

1.1092 

0.81 

0.82 

0.915034 

1.355466 

0.67507 

1.4813 

0-7377 

1.0928 

0.82 

0.83 

0.928635 

1.364684 

0.68047 

1.4696 

0.7327 

1.0768 

0.83 

0.84 

0.942328 

I-374039 

0.68580 

1.4582 

0.7278. 

1.0612 

0.84 

0.85 

0.956116 

1-383531 

0.69107 

1.4470 

0.7228 

1.0459 

0.85 

0.86 

0.969999 

1.393161 

0.69626 

1-4362 

0.7178 

1.0309 

0.86 

0.87 

0.983980 

1.402931 

0.70137 

1.4258 

0.7128 

1.0163 

0.87 

0.88 

0.998058 

1.412841 

0.70642 

1.4156 

0.7078 

I.OO2O 

0.88 

0.89 

1.012237 

1.422893 

0.71139 

1-4057 

0.7028 

0.9881 

0.89 

0.90 

1.026517 

1433086 

0.71629 

1.3961 

0.6978 

0-9737 

0.90 

0.91 

1.040899 

1.443423 

0.72114 

1.3867 

0.6928 

0.9607 

0.91 

0.92 

1.055386 

I-453905 

0.72591 

1.3776 

0.6878 

0-9475 

0.92 

0-93 

1.069978 

1.464531 

0.73060 

1-3687 

0.6828 

0.9346 

o-93 

0.94 

1.084677 

1.475305 

0.73522 

1.3600 

0.6778 

0.9219 

0.94 

o-95 

1.099484 

1.486225 

0-73979 

i.35i8 

0.6728 

0.9095 

0-95 

0.96 

1.114402 

I-497295 

0.74427 

1.3436 

0.6678 

0.8973 

0.96 

0.97 

1.129431 

1.508514 

0.74870 

1.3356 

0.6629 

0.8854 

0.97 

0.98 

I-I44573 

1.519884 

0.75306 

1.3279 

0.6579 

0.8737 

0.98 

0.99 

1.159829 

1.531406 

0.75736 

1.3204 

0.6529 

0.8621 

0-99 

Example.      cosh  0.55  =  1.155101. 


[i45] 


TABLE  XV 
REAL  HYPERBOLIC  FUNCTIONS.    /  (x  +  io)  =  u  +  io.     CONTINUED 


e 

Sinh0 

Coshfl 

Tanh0 

Cothfl 

Sech0 

Cosech  6 

e 

1.  00 

1.175201 

1.543081 

0.76159 

1-3130 

0.6480 

0.8509 

I.OO 

I.OI 

1.190691 

1.554910 

0.76576 

1-3059 

0.6431 

0.8395 

I.OI 

1.02 

1.206300 

1.566895 

0.76987 

1.2989 

0.6382 

0.8290 

1.02 

1.03 

1.222029 

I-579036 

0.77391 

1.2921 

0.6333 

0.8183 

1-03 

1.04 

1.237881 

i-59I336 

0.77789 

1.2855 

0.6284 

0.8078 

I.O4 

1.05 

1.253857 

1.603794 

0.78181 

1.2791 

0.6235 

0-7975 

1.05 

1.  06 

1.269958 

1.616413 

0.78566 

1.2728 

0.6  1  86 

0.7874 

1.  06 

1.07 

1.286185 

1.629194 

0.78946 

1.2666 

0.6138 

0.7777 

1.07 

1.  08 

1.302542 

1.642138 

0.79320 

1.2607 

0.6090 

0.7677 

1.  08 

1.09 

1.319029 

1-655245 

0.79688 

1-2549 

0.6042 

0.7581 

I.O9 

I.IO 

I-33S647 

1.668519 

0.80050 

1.2492 

0-5993 

0.7487 

I.IO 

I.  II 

1.352400 

1.681959 

0.80406 

1-2437 

0-5945 

0-7393 

I.  II 

I.I2 

1.369287 

1-695567 

0.80757 

1.2382 

0.5898 

0.7302 

1.  12 

I-I3 

1.386312 

1.709345 

0.81102 

1.2330 

0.5850 

0.7215 

I-!3 

I.I4 

I-403475 

1.723294 

0.81441 

1.2279 

0.5803 

0.7125 

1.14 

I-IS 

1.420778 

I.7374I5 

0.81775 

1.2229 

0-5755 

0.7038 

1-15 

1.16 

1.438224 

1.751710 

0.82104 

i.  2180 

0.5708 

0.6953 

1.16 

1.17 

1-455813 

1.766180 

0.82427 

1.2132 

0.5662 

0.6869 

1.17 

1.18 

1.473548 

1.780826 

0.82745 

1.2085 

0.5616 

0.6786 

1.18 

1.19 

1.491430 

I.79565I 

0.83058 

1.2040 

0-5569 

0.6705 

1.19 

1.20 

1.509461 

1.810656 

0.83365 

I-I995 

0-5523 

0.6625 

i.  20 

I.2I 

1.527644 

1.825841 

0.83668 

1.1952 

0-5477 

0.6546 

1.  21 

1.22 

1-545979 

1.841209 

0.83965 

1.1910 

o.543i 

0.6468 

1.22 

1.23 

1.564468 

1.856761 

0.84258 

1.1868 

0.5385 

0.6392 

1.23 

1.24 

i.583"5 

1.872499 

0.84546 

1.1828 

0-5340 

0.6317 

1.24 

I.2S 

1.601919 

1.888424 

0.84828 

1.1789 

0.5296 

0.6242 

1-25 

1.26 

1.620884 

1.904538 

0.85106 

1.1750 

0.5251 

0.6170 

1.26 

1.27 

1.640010 

1.920842 

0.85380 

1.1712 

0.5206 

0.6098 

1.27 

1.28 

1.659301 

1-937339 

0.85648 

1.1675 

0.5162 

0.6026 

1.28 

I.2Q 

1.678758 

1.954029 

0.85913 

1.1640 

0.5118 

0-5957 

1.29 

1.30 

1.698382 

1.970914 

0.86172 

1.1605 

0.5074 

0.5888 

1.30 

I-3I 

1.718177 

1.987997 

0.86428 

1.1570 

0.5030 

0.5820 

I-3I 

1.32 

1.738143 

2.005278 

0.86678 

i-i537 

0.4987 

0-5753 

1.32 

1-33 

1.758283 

2.022760 

0.86925 

1.1504 

0.4944 

0.5687 

i-33 

1-34 

I-778599 

2.040445 

0.87167 

1.1472 

0.4901 

0.5623 

i-34 

I-3S 

1.799093 

2-058333 

0.87405 

1.1441 

0.4858 

0-5559 

i-35 

1.36 

1.819766 

2.076427 

0.87639 

1.1410 

0.4816 

0-5495 

1-36 

1-37 

1.840622 

2.094729 

0.87869 

1.1380 

0-4773 

0-5433 

i-37 

1.38 

1.861662 

2.113240 

0.88095 

L^Si 

0.4732 

0.5372 

1-38 

1-39 

1.882887 

2.131963 

0.88317 

1-1323 

0.4690 

0.53H 

1-39 

1.40 

1.904302 

2.150898 

0.88535 

1.1295 

0.4649 

0.5252 

1.40 

1.41 

1.925906 

2.170049 

0.88749 

1.1268 

0.4608 

0.5192 

1.41 

1.42 

L947703 

2.189417 

0.88960 

1.1241 

0.4568 

o.5i34 

1.42 

1-43 

1.969695 

2.209004 

0.89167 

1.1215 

0.4527 

0.5077 

i-43 

1.44 

1.991884 

2.228812 

0.89370 

1.1189 

0.4486 

0.5020 

1.44 

i.  45 

2.014272 

2.248842 

0.89569 

1.1165 

0.4446 

0.4964 

1-45 

1.46 

2.036862 

2.269098 

0.89765 

1.1140 

0.4407 

0.4909 

1.46 

1.47 

2-059655 

2.289580 

0.89958 

1.1116 

0.4367 

0.4855 

1-47 

1.48 

2.082654 

2.310292 

0.90147 

1.1093 

0.4329 

0.4802 

1.48 

1.49 

2.105861 

2-331234 

0.90332 

1.1070 

0.4290 

0.4749 

1.49 

Example.      tanh  1.25  =  0.84828. 
[146] 


TABLE  XV 
REAL  HYPERBOLIC  FUNCTIONS.    /  (x  +  io)  =  u  +  to.     CONTINUED 


e 


Sinh0 


Cosh0        Tanh0       Coth  0      Sech  6     Cosechfl      6 


1.50 

2.129279 

2.352410 

0.90515 

1.1048 

0.4251 

0.4697 

1.50 

1-51 

2.152910 

2.373820 

0.90694 

1.1026 

0.4212 

0.4645 

i-Si 

1-52 

2.176757 

2.395469 

0.90870 

1.1005 

0.4174 

0.4594 

1.52 

1-53 

2.200821 

2.417356 

0.91042 

1.0984 

0.4137 

0-4543 

1-53 

1-54 

2.225105 

2.439486 

0.91212 

1.0963 

0.4099 

0.4494 

1-54 

i-SS 

2.249611 

2.461859 

0.91379 

1.0944 

0.4062 

0.4444 

1-55 

1.56 

2-274343 

2.484479 

0.91542 

1.0924 

0.4025 

0.4398 

1.56 

1-57 

2.299302 

2.507347 

0.91703 

1.0905 

0.3988 

0.4350 

1-57 

1.58 

2.324490 

2-530465 

0.91860 

1.0886 

0-3952 

0.4302 

1.58 

1-59 

2.349912 

2.553837 

0.92015 

1.0868 

0.3916 

0.4255 

1-59 

i.  60 

2-37SS68 

2.577464 

0.92167 

1.0850 

0.3879 

0.4209 

1.60 

1.61 

2.401462 

2.601349 

0.92316 

1.0832 

0.3844 

0.4164 

1.61 

1.62 

2.427596 

2.625495 

0.92462 

1.0815 

0.3809 

0.4119 

1.62 

1.63 

2-453973 

2.649902 

0.92606 

1.0798 

0-3774 

0.4075 

1.63 

1.64 

2.480595 

2.674575 

0.92747 

1.0782 

0-3739 

0.4031 

1.64 

1-65 

2-507465 

2-699515 

0.92886 

1.0766 

0.3704 

0.3988 

1.65 

1.66 

2.534586 

2.724725 

0.93022 

1.0750 

0.3670 

0-3945 

1.66 

1.67 

2.561960 

2.750207 

0.93155 

1.0735 

0.3636 

0.3903 

1.67 

1.68 

2-589591 

2-775965 

0.93286 

,  1.0719 

0.3602 

0.3862 

1.68 

1.69 

2.617481 

2.802000 

0.93415 

1.0704 

0-3569 

0.3820 

1.69 

1.70 

2.645632 

2.828315 

0-93541 

1.0691 

0.3536 

0.3780 

1.70 

1.71 

2.674048 

2.854914 

0.93665 

1.0676 

0-3503 

0.3740 

1.71 

1.72 

2.702731 

2.881797 

0.93786 

1.0662 

0.3470 

0.3700 

1.72 

1-73 

2.731685 

2.908969 

0.93906 

1.0649 

0.3438 

0.3661 

1-73 

1.74 

2.760912 

2.936432 

0.94023 

1.0636 

0.3405 

0.3622 

1.74 

i-75 

2.790414 

2.964188 

0.94138 

1.0623 

0-3373 

0.3584 

i-75 

1.76 

2.820196 

2.992241 

0.94250 

1.0610 

0.3342 

0.3546 

1.76 

1.77 

2.850260 

3.020593 

0.94361 

1-0597 

0.3310 

0.3508 

1.77 

1.78 

2.880609 

3.049247 

0.94470 

1-0585 

0.3279 

o.347i 

1.78 

1.79 

2.911246 

3.078206 

0.94576 

1-0573 

0.3248 

0-3435 

1.79 

1.80 

2.942174 

3-107473 

0.94681 

1.0562 

0.3218 

0-3399 

1.80 

1.81 

2-973397 

3-137051 

0.94783 

1-0550 

0.3187 

0.3363 

1.81 

1.82 

3.004916 

3.166942 

0.94884 

1.0539 

0.3158 

0.3328 

1.82 

1-83 

3-036737 

3-r97i5o 

0.94983 

1.0528 

0.3128 

0.3293 

1.83 

1.84 

3.068860 

3.227678 

0.95080 

1.0517 

0.3098 

0.3258 

1.84 

1.85 

3.101291 

3-258528 

0.95175 

1.0507 

0.3069 

0.3224 

1.85 

1.86 

3.134032 

3.289705 

0.95268 

1.0497 

0.3040 

0.3191 

1.86 

1.87 

3.167086 

3.321210 

0-95359 

1.0487 

0.3011 

0.3157 

1.87 

1.88 

3.200457 

3-353047 

0.95449 

1.0477 

0.2982 

0-3125 

1.88 

1.89 

3.234148 

3.385220 

0-95537 

1.0467 

0.2954 

0.3092 

1.89 

1.90 

3.268163 

3-4I7732 

0.95624 

1.0458 

0.2926 

0.3059 

1.90 

1.91 

3.302504 

3-450585 

0.95709 

1.0448 

0.2897 

0.3028 

1.91 

1.92 

3-337I76 

3-483783 

0.95792 

1.0439 

0.2870 

0.2997 

1.92 

i-93 

3.372181 

3-5I7329 

0.95873 

1.0430 

0.2843 

0.2965 

1-93 

1.94 

3.407524 

3-55I227 

0-95953 

1.0422 

0.2816 

0.2935 

1.94 

I-  95 

3.443207 

3-58548i 

0.96032 

1.0413 

0.2789 

0.2904 

i-95 

1.96 

3.479234 

3.620093 

0.96109 

1.0405 

0.2762 

0.2874 

1.96 

1.97 

3.515610 

3-655067 

0.96185 

1.0397 

0.2736 

0.2844 

1.97 

1.98 

3-552337 

3.690406 

0.96259 

1.0389 

0.2710 

0.2815 

1.98 

1.99 

3-5894I9 

3-726115 

0.96331 

1.0380 

0.2684 

0.2786 

1.99 

Example.       coth  1.70  =  1.0691. 


[147] 


TABLE  XV 
REAL  HYPERBOLIC  FUNCTIONS,    f  (x  +  io)  =  u  +  io.     CONTINUED 


e 

Sinh0 

Cosh0 

Tanhfl 

Cothfl 

Sechfl 

Cosech  0 

0 

2.OO 

3.626860 

3.762196 

0.96403 

J-0373 

0.2658 

0.2757 

2.00 

2.OI 

3.66466 

3-79865 

0.96473 

1-0365 

0.2632 

0.2729 

2.OI 

2.O2 

3.70283 

3-83549 

0.96541 

1.0358 

0.2607 

0.2701 

2.O2 

2.03 

3-74I38 

3.87271 

0.96608 

1-0351 

0.2582 

0.2673 

2.03 

2.04 

3.78029 

3.91032 

0.96675 

1.0344 

0.2557 

0.2645 

2.04 

2.05 

3-81958 

3-94832 

0.96740 

1-0337 

0-2533 

0.2618 

2.05 

2.O6 

3.85926 

3.98671 

0.96803 

1.0330 

0.2508 

0.2596 

2.06 

2.07 

4.02550 

0.96865 

1.0323 

0.2484 

0.2565 

2.07 

2.08 

3-93977 

4.06470 

0.96926 

1.0317 

0.2460 

0.2538 

2.08 

2.O9 

3.98061 

4.10430 

0.96986 

1.0310 

0.2436 

0.2512 

2.09 

2.IO 

4.02  1£6 

4.14431 

0.97045 

1.0305 

0.2413 

0.2486 

2.IO 

2.  II 

4.06350 

4.18474 

0.97103 

1.0298 

0.2389 

0.2461 

2.  II 

2.12 

4-10555 

4-22558 

0.97159 

1.0293 

0.2366 

0.2436 

2.12 

2.13 

4.14801 

4.26685 

0.97215 

1.0286 

0.2344 

0.2411 

2.13 

2.14 

4.19089 

4-30855 

0.97269 

1.0280 

0.2321 

0.2386 

2.14 

2.15 

4.23419 

4-35067 

0.97323 

1.0275 

0.2298 

0.2362 

2-15 

2.l6 

4.27791 

4.39323 

0-97375 

1.0269 

0.2276 

0.2338 

2.l6 

2.17 

4.32205 

4-43623 

0.97426 

1.0264 

0.2254 

0.2314 

2.17 

2.18 

4.36663 

447967 

0.97477 

1.0259 

0.2232 

0.2290 

2.18 

2.19 

4.41165 

4.52356 

0.97526 

1.0254 

O.22II 

0.2267 

2.19 

2.20 

4-457" 

4.56791 

0-97574 

1.0249 

0.2189 

0.2244 

2.  2O 

2.21 

4.50301 

4.61271 

0.97622 

1.0243 

0.2168 

0.2221 

2.21 

2.22 

4-54936 

4-65797 

0.97668 

1.0239 

0.2147 

0.2198 

2.22 

2.23 

4.59617 

4.70370 

0.97714 

1.0234 

O.2I26 

O.2I76 

2.23 

2.24 

4.64344 

4.74989 

0.97758 

1.0229 

0.2105 

0.2154 

2.24 

2.25 

4.69117 

4.79657 

0.97803 

1.0225 

0.2085 

O.2I32 

2.25 

2.26 

4-73937 

4.84372 

0.97847 

I.022O 

0.2064 

O.2IIO 

2.26 

2.27 

4.78804 

4.89136 

0.97888 

I.  O2  1  6 

0.2044 

0.2089 

2.27 

2.28 

4.83720 

4.93948 

0.97929 

I.O2II 

O.2024 

O.2067 

2.28 

2.29 

4.88683 

4.98810 

0.97970 

I.O2O7 

0.2OO5 

O.2047 

2.29 

2.30 

4.93696 

5.03722 

0.98010 

1.0203 

0.1985 

O.2O26 

2.30 

2.31 

4.98758 

5.08684 

0.98049 

I.OI99 

0.1966 

0.2005 

2.31 

2.32 

5.03870 

5^3697 

0.98087 

I.OI95 

0.1947 

0.1985 

2.32 

2-33 

5.09032 

5.18762 

0.98124 

I.Oigi 

0.1928 

0.1965 

2-33 

2-34 

5^4245 

5-23879 

0.98161 

I.OI87 

0.1909 

0.1945 

2-34 

2-35 

5.19510 

5.29047 

0.98198 

I.Ol84 

0.1890 

0.1925 

2-35 

2.36 

5-24827 

5-34269 

0.98233 

I.OlSo 

0.1872 

0.1905 

2.36 

2-37 

5.30196 

5-39544 

0.98268 

I.OI77 

0.1854 

0.1886 

2-37 

2.38 

5.35618 

5-44873 

0.98302 

I.OI73 

0.1835 

0.1867 

2.38 

2-39 

5-41093 

5-50256 

0.98335 

1.0169 

O.l8l7 

0.1848 

2-39 

2.40 

5.46623 

5-55695 

0.98368 

1.0166 

0.1800 

0.1829 

2.40 

2.41 

5.52207 

5.611-89 

0.98399 

1.0163 

0.1782 

0.1811 

2.41 

2.42 

5-57847 

5-66739 

0.98431 

1.0159 

0.1765 

0.1793 

2.42 

2-43 

5-63542 

5-72346 

0.98462 

1.0156 

0.1747 

0.1775 

2-43 

2-44 

5.69294 

5.78010 

0.98492 

1-0153 

0.1730 

0.1757 

2-44 

2-45 

5-7SI03 

5.83732 

0.98522 

1.0150 

O.I7I3 

0.1739 

2-45 

2.46 

5.80969 

5.89512 

0.98551 

1.0147 

0.1696 

0.1721 

2.46 

2-47 

5.86893 

5-95352 

0.98579 

1.0144 

o.i  680 

0.1704 

2-47 

2.48 

5.92876 

6.01250 

0.98607 

1.0141 

0.1663 

0.1687 

2.48 

2.49 

5-98918 

6.07209 

0.98635 

1.0138 

0.1647 

0.1670 

2.49 

Example. 

sech  2.00  =  0.2658. 

[148] 

TABLE  XV 
REAL  HYPERBOLIC  FUNCTIONS.    /  (x  +  to)  =  u  +  io.     CONTINTJED 


0 

Sinhfl 

Coshfl 

Tanh0 

Cothfl 

Sech0 

Cosech  8 

6 

*-5 

6.05020 

6.13229 

0.98661 

1.0136 

0.1631 

0.1653 

2-5 

2.6 

6.69473 

6.76901 

0.98903 

I.OIII 

0.1477 

0.1494 

2.6 

2.7 

7.40626 

7-47347 

0.99101 

1.0091 

0.1338 

0.1350 

2.7 

2.8 

8.19192 

8.25273 

0.99263 

1.0074 

O.I2I2 

O.I22I 

2.8 

2.9 

9-05956 

9.11458 

0.99396 

1.  0061 

0.1097 

0.1104 

2.9 

3-o 

10.01787 

10.06766 

0-99505 

1.0050 

0.0937 

0.09982 

3-o 

3-i 

11.07645 

11.12150 

0-99595 

1.0041 

0.0899 

0.0003 

3-i 

3-2 

12.24588 

12.28665 

0.99668 

1.0033 

0.0814 

0.08  1  6 

3-2 

3-3 

13.53788 

I3-57476 

0.99728 

1.0027 

0.0736 

0.0739 

3-3 

3-4 

14.96536 

14.99874 

0.99778 

1.0022 

0.0667 

0.0668 

3-4 

3-5 

16.54263 

16.57282 

0.99818 

1.0018 

0.0604 

0.0604 

3-S 

3-6 

18.28546 

18.31278 

0.99851 

1.0015 

0.0546 

0.0547 

3.6 

3-7 

20.21129 

20.23601 

0.99878 

I.OOI2 

0.0494 

0.0495 

3-7 

3-8 

22.33941 

22.36178 

0.99900 

I.OOIO 

0.0447 

0.0448 

3-8 

3-9 

24.69110 

24.7"3S 

0.99918 

1.0008 

0.0405 

0.0405 

3-9 

4.0 

27.28992 

27.30823 

0-99933 

1.0007 

0.0366 

0.0366 

4.0 

4.1 

30.16186 

30.17843 

0-99945 

1.  0006 

0.0331 

0.0332 

4-1 

4-2 

33-33567 

33-35066 

0-99955 

1.0005 

0.0300 

0.0300 

4.2 

4-3 

36.84311 

36.85668 

0.99963 

1.0004 

0.0271 

0.0271 

4-3 

4.4 

40.71930 

40.73157 

0.99970 

1.0003 

0.0245 

0.0245 

4-4 

4-5 

45.00301 

45.01412 

0-99975 

1.0003 

O.O222 

O.O222 

4-5 

4.6 

49-737I3 

49.74718 

0.99980 

I.OOO2 

O.O2OI 

O.O2OI 

4.6 

4-7 

54.96904 

54.97813 

0.99983 

I.OO02 

0.0182 

0.0182 

4-7 

4-8 

60.75109 

60.75932 

0.99986 

I.OOOI 

0.0165 

0.0165 

4.8 

4-9 

67.14117 

67.14861 

0.99989 

1.  0001 

0.0149 

0.0149 

4-9 

S-° 

74.20321 

74.20995 

0.99991 

I.OOOI 

0.0135 

0.0135 

5-o 

5-1 

82.0079 

82.0140 

0-99993 

1.00007 

0.01219 

0.01219 

5-i 

5-2 

90.6334 

90.6389 

0.99993 

1.00007 

0.01103 

0.01103 

S-* 

5-3 

100.1659 

100.1709 

0-99994 

1.00006 

0.00998 

0.00998 

5-3 

5-4 

110.7009 

110.7055 

0.99995 

1.00005 

0.00903 

0.00903 

5-4 

5-5 

122.3439 

122.3480 

0.99996 

1.00004 

0.00818 

0.00818 

5-5 

5-6 

135.2114 

135.2150 

0-99997 

1.00003 

0.00740 

0.00740 

5-6 

5-7 

149.4320 

J49-4354 

0.99998 

I.OOO02 

0.00669 

0.00669 

5-7 

5-8 

165.1483 

165.1513 

0.99998 

I.OOOO2 

0.00606 

0.00606 

5-8 

5-9 

182.5174 

182.5201 

0.99998 

1.00002 

0.00548 

0.00548 

5-9 

6.0 

201.7132 

201.7156 

0-99999 

I.OOOOI 

0.00496 

0.00496 

6.0 

6.1 

222.9278 

222.9300 

I.OOO 

I.OOO 

0.00449 

0.00449 

6.1 

6.2 

246.3735 

246.3755 

I.OOO 

I.OOO 

0.00406 

0.00406 

6.2 

6-3 

272.2850 

272.2869 

I.OOO 

I.OOO 

0.00367 

0.00367 

6-3 

6.4 

300.9217 

300.9233 

I.OOO 

I.OOO 

0.00332 

0.00332 

6.4 

6-5 

332.5701 

332.5716 

I.OOO 

I.OOO 

0.00301 

0.00301 

6-5 

6.6 

367.5469 

367-5483 

I.OOO 

I.OOO 

0.00272 

0.00272 

6.6 

6-7 

406.2023 

406.2035 

I.OOO 

I.OOO 

0.00246 

0.00246 

6.7 

6.8 

448.9231 

448.9242 

I.OOO 

I.OOO 

0.00223 

0.00223 

6.8 

6.9 

496.1369 

496.1379 

I.OOO 

I.OOO 

O.OO2O2 

0.00202 

6.9 

7.0 

548.3161 

548.3170 

I.OOO 

I.OOO 

0.00182 

0.00182 

7-o 

7-i 

605.9831 

605.9839 

I.OOO 

I.OOO 

0.00165 

0.00165 

7-i 

7-2 

669.7150 

669.7158 

I.OOO 

I.OOO 

0.00149 

0.00149 

7.2 

7-3 

740.1496 

740.1503 

I.OOO 

I.OOO 

0.00135 

0.00135 

7-3 

7-4 

817.9919 

817.9925 

I.OOO 

I.OOO 

0.00122 

O.OOI22 

7-4 

7-5 

904.0209 

904.0215 

I.OOO 

I.OOO 

O.OOIII 

O.OOI  1  1 

7-S 

Example.      cosech  2.50  =  0.1653. 
[149] 


TABLE  XVI.     SUBDIVISIONS  OF  A  DEGREE  —  AUXILIARY  TABLE 


O.OI 

00.36 

O.4I 

24.36 

0.81 

48.36 

OI 

0.0167  • 

4i 

0.6833 

21 

0.0058 

O.O2 

OI.I2 

0.42 

25.12 

0.82 

49.12 

02 

0.0333 

42 

O.7OOO 

22 

0.006  1 

0.03 

01.48 

0-43 

25.48 

0.83 

49.48 

03 

0.0500 

43 

0.7167 

23 

0.0064 

O.O4 

02.24 

0.44 

26.24 

0.84 

50.24 

04 

0.0667 

44 

0.7333 

24 

0.0067 

O.OS 

03.00 

o-4S 

27.00 

0.85 

51.00 

05 

0.0833 

45 

0.7500 

25 

0.0069 

O.O6 

03-36 

0.46 

27-36 

0.86 

51.36 

06 

O.IOOO 

46 

0.7667 

26 

0.0072 

O.O7 

O4.I2 

0.47 

28.12 

0.87 

52.12 

07 

0.1167 

47 

0.7833 

27 

0.0075 

0.08 

04.48 

0.48 

28.48 

0.88 

52.48 

08 

0.1333 

48 

O.SOOO 

28 

0.0078 

O.O9 

05.24 

0.49 

29.24 

0.89 

53-24 

09 

0.1500 

49 

0.8167 

29 

0.0081 

O.IO 

O6.OO 

0.50 

30.00 

0.90 

54-oo 

IO 

0.1667 

So 

0.8333 

30 

0.0083 

O.I  I 

06.36 

0.51 

30-36 

0.91 

54.36 

II 

0.1833 

Si 

0.8500 

31 

0.0086 

O.I2 

07.12 

0.52 

31.12 

0.92 

5S-12 

12 

0.2OOO 

52 

0.8667 

32 

0.0089 

0.13 

07.48 

0-53 

31.48 

o-93 

55-48 

13 

0.2167 

53 

0.8833 

33 

0.0092 

O.I4 

08.24 

0-54 

32.24 

0.94 

56-24 

14 

0.2333 

54 

0.9OOO 

34 

0.0094 

O.IS 

09.00 

o-SS 

33-00 

0-95 

57.00 

15 

0.2500 

55 

0.9167 

35 

0.0097 

0.16 

09.36 

0.56 

33-36 

0.96 

57-36 

16 

0.2667 

56 

0-9333 

36 

O.OIOO 

0.17 

IO.I2 

0-57 

34-12 

0.97 

58.12 

17 

0.2833 

57 

0.9500 

37 

0.0103 

0.18 

10.48 

0.58 

34-48 

0.98 

58.48 

18 

0.3000 

58 

0.9667 

38 

0.0106 

0.19 

11.24 

o-S9 

35-24 

0.99 

59-24 

19 

0.3167 

59 

0-9833 

39 

0.0108 

0.20 

12.00 

0.60 

36.00 

I.OO 

60.00 

20 

0-3333 

60 

I.OOOO 

40 

O.OI  1  1 

a 

0 

O.2I 

12.36 

0.61 

36.36 

21 

O.3SOO 

OI 

0.0003 

41 

0.0114 

O.22 

I-J.I2 

0.62 

OX/"»JV' 
37.12 

22 

oo^^ 
0.3667 

02 

0.0006 

42 

0.0117 

O.23 

*O*    * 

IV48 

0.63 

o  /  •  *•* 
27.48 

23 

v"O'-'v-'  / 

0.3833 

03 

0.0008 

43 

0.0119 

^•"O 

O.24 

J.  J.4J.V 
14.24 

*•"  "O 

0.64 

O  1  *^** 

38.24 

*j 
24 

•J     JO 

0.4000 

o 
°4 

O.OOII 

~O 

44 

O.OI  2  2 

o.2<; 

I^.OO 

0.6? 

O"   *^ 

30.00 

2? 

0.4167 

oc 

0.0014 

41; 

O.OI25 

•*•*•  j 
0.26 

*  J**" 

1C.  76 

"•"o 
0.66 

o  y 
30.36 

~«j 
26 

O.4333 

J 

06 

0.0017 

~j 
46 

O.OI28 

0.27 

O   O 

16.12 

0.67 

ov  o 

40.12 

•  •  •  • 

•  •  .  « 

27 

V.*TOOO 
0.4500 

07 

0.0019 

47 

O.OI3I 

0.28 

16.48 

0.68 

40.48 

28 

O.4.66? 

08 

O.OO22 

48 

O.O1  11 

O.2Q 

.  vs.tfi_r 

17.24 

0.60 

*T        ^ 

4.1.24. 

2O 

v.lfwu  y 
O.4833 

oo 

O.OO25 

•f** 

4O 

•WAOi> 

0.0136 

y 
O.1O 

•  /  •  *"T 

iS.OO 

'     y 
O.7O 

^*  •  *^ 
42.OO 

*y 
30 

^•t^OO 

0.5000 

vy 

IO 

O.OO28 

ty 

CO 

O.OI  3O 

*~*o 

**•  f  >* 

T^ 

o 

/     // 

ow 

0 

-WAoy 

0.31 

18.36 

0.71 

42.36 

0.001 

00.03.6 

3i 

0.5167 

II 

0.0031 

51 

0.0142 

0.32 

19.12 

0.72 

43.12 

O.O02 

00.07.2 

32 

0-5333 

12 

0.0033 

52 

0.0144 

o-33 

19.48 

o-73 

43-48 

0.003 

06.10.8 

33 

0.5500 

13 

0.0036 

53 

0.0147 

0-34 

20.24 

0.74 

44.24 

0.004 

00.14.4 

34 

0.5667 

14 

0.0039 

54 

0.0150 

0-35 

21.00 

0-75 

45-00 

0.005 

00.18 

35 

0-5833 

15 

O.OO42 

55 

0.0153 

0.36 

21.36 

0.76 

45-36 

0.006 

OO.2I.6 

36 

0.6000 

16 

O.OO44 

56 

0.0156 

0-37 

22.12 

0.77 

46.12 

0.007 

00.25.2 

37 

0.6167 

17 

O.0047 

57 

0.0158 

0.38 

22.48 

0.78 

46.48 

0.008 

00.28.8 

38 

0.6333 

18 

O.OO5O 

58 

0.0161 

o-39 

23.24 

0.79 

47.24 

0.009 

00.32.4 

39 

0.6500 

19 

0.0053 

59 

0.0164 

0.40 

24.00 

0.80 

48.00 

O.OIO 

00.36 

40 

0.6667 

20 

0.0055 

60 

0.0167 

Examples.      o°.4i    =  o°.24'.j6" 
o°.oo5  =  o°.oo'.i8* 


o°.4i'.oo*  =  o°.6833. 
o°.oo'.46*  =  o°.oi28. 


[150] 


EXPLANATORY   TEXT 


EXPLANATORY    TEXT 

INTRODUCTION 

THE  Tables  in  this  book  are  designed  primarily  for  presenting  hyperbolic  functions 
of  a  complex  variable  either  in  the  rectangular  coordinate  form  of  that  variable  (x  -f-  iy) 
or  the  polar  coordinate  form  (p/8).  They  are  also  designed  secondarily  for  presenting 
circular  functions  of  a  complex  variable.  A  few  formulas  are  added  as  aids  to  the 
conversion  of  such  functions.  The  most  extensive  range  offered  is  in  Tables  VII  to 
XIV  inclusive,  between  which,  the  functions  sinh  (x  +  iy),  cosh  (x  +  iy),  tanh  (x  -f  iy), 
expressed  in  the  result  either  in  rectangular  coordinates  u  +  iv  or  in  polar  coordinate 
quantities  r  /y,  may  be  obtained  between  the  limits  of  o  and  ±  10  of  x,  and  between  the 
limits  of  o  and  ±  «  for  y.  It  is  shown,  moreover,  to  be  an  easy  matter  to  extend  the 
range  of  x  beyond  the  offered  range  of  ±  10,  should  such  an  extension  be  required. 
The  practical  need  for  tabulated  values  of  hyperbolic  functions  of  (x  +  iy)  beyond  the 
range  of  x  =  ±10  appears  to  be  so  small  that  any  such  extension  is  left  to  the  reader. 

As  the  author's  applications  for  financial  assistance  in  the  computation  of  the  Tables 
were  unsuccessful,  the  steps  in  x  and  y  (0.05  and  0.07854  respectively)  are  larger  than 
were  originally  intended;  i.e.,  for  reducing  the  work  of  the  user  to  the  lowest  practic- 
able limits.  Consequently,  interpolation  must  ordinarily  be  resorted  to,  when  three  or 
more  significant  digits  are  needed  in  the  results.  Such  interpolations  require  an  appre- 
ciable amount  of  time  to  effect  in  two  dimensions;  i.e.,  for  both  x  and  y.  In  order  to 
render  such  interpolation  unnecessary  for  ordinary  engineering  purposes,  where  three,  or 
at  most  four,  significant  digits  may  be  needed,  a  separate  atlas  of  23  large-scale  charts, 
45  cm.  X  45  cm.  over  ruled  areas,  has  been  prepared,  and  is  published  as  an  adjunct 
to  these  Tables.  The  necessary  interpolation  can  very  swiftly  be  made  on  the  charts, 
by  inspection. 

COMPLEX   QUANTITIES 

The  following  brief  outline  of  complex  quantities  is  offered  in  view  of  their  funda- 
mental importance  in  connection  with  the  Tables,  for  the  assistance  of  those  who  have 
studied  elementary  mathematics,  but  who  may  not  have  become  familiar  with  com- 
plex numbers.  For  a  more  comprehensive  discussion  of  complex  quantities,  the  reader 
must  be  referred  to  special  treatises  on  the  subject. 

Ordinary  numerical  quantities,  or  the  numbers  dealt  with  in  ordinary  arithmetic, 
may  be  considered  to  range  between  zero  and  either  positive  or  negative  infinity,  by 
indefinitely  small  gradations.  Such  numbers  may  be  represented  geometrically  by 
distances,  in  either  direction,  from  a  zero  point  on  an  infinite  straight  line.  Thus  in 
Fig.  i,  we  may  consider  that  the  straight  line  —  XOX  extends  from  minus  infinity  on 
the  left,  to  plus  infinity  on  the  right,  O  being  the  zero  point.  The  point  x\.  would  then 
represent  +  i,  and  so  on.  That  is,  the  number  +  i  may  be  regarded  as  represented 
on  the  line  —  XOX  either  by  the  position  of  the  point  xi  with  respect  to  the  zero  point  0; 

[153] 


EXPLANATORY  TEXT 

or,  as  the  vector  Ox\;  i  e.,  the  straight  line  drawn  from  the  origin  O  to  the  point  x\  and 
forming  a  part  of  the  reference  line  —  XOX.  Under  these  assumptions,  the  ordinary 
numbers  of  arithmetic  may  be  represented  geometrically  as  vectors;  but  such  vectors 
are  confined  to  a  single  straight-line  direction  from  O  towards  .X"  for  positive  numbers, 
and  from  O  towards  —  X  for  negative  numbers. 

Complex  quantities,  or  complex  numbers,  cannot  be  completely  represented  by 
reference  to  a  single  direction,  or  to  vectors  along  one  and  the  same  straight  line.  They 
may,  however,  be  represented  geometrically  by  the  position,  in  an  infinite  plane,  of  a 
movable  point  with  respect  to  a  fixed  point  as  origin.  Thus,  in  Fig.  i,  the  plane  XOY 
is  the  plane  of  reference,  and  the  fixed  point  0  is  the  origin.  Then  any  point  PI  in  the 
plane  represents  a  complex  number,  and  any  complex  number  may  be  represented  by 
a  point  on  the  plane. 


-Y 


2  X 


FIG.  2.  —  Plane  Vector  2.236  e"-106  or 


FIG.  i.  —  Complex  quantity  i  +  »  a.  designated  by  2.236  763°. 26'. 

A  complex  number  may  be  specified  either  in  rectangular  coordinates,  or  in  polar 
coordinates,  as  may  be  preferred.  Thus,  the  same  vector  OP\  is  represented  in  Fig.  i 
to  rectangular  coordinates,  and  in  Fig.  2  to  polar  coordinates.  In  Fig.  i,  the  X  axis 

—  XOX  passing  through  the  origin  O  is  the  fundamental  reference  axis,  and  the  Y  axis 

—  YOY,  perpendicular  thereto  in  the  reference  plane,  immediately  follows.     Then  the 
point  Pi,  measuring  +  i  along  OX,  and  -f-  2  along  OF,  may  be  defined  by  the  expres- 
sion (i  +  22),  where  the  symbol  i  signifies  measurement  along  'the  subordinate  axis. 
It  is  shown  in  mathematical  treatises  that  *  =  V  —  i.      The  vector  OP\  of  Fig.  i  may 
therefore  be  expressed  as  (i  +  V~  1-2)  and  a  vector  from  0  to  any  point  in  the  plane 
may  be  represented  by  x  +  V~i  y  —  x  +  *3S  where  x  and  y  may  have  any  positive  or 
negative  numerical  values,  including  zero. 

In  pursuance  of  time-honored  terminology,  the  axis  —  XOX  is  sometimes  called  the 
"real"  axis,  and  —  YOY  the  "imaginary,"  axis;  so  that  the  ^-component  of  a 
complex  number  becomes  the  "  real  component,"  and  the  ^-component  the  "  imaginary 
component."  The  symbol  i  still  stands  for  the  imaginary  component.  In  mathematics 
as  applied  to  electrical  engineering,  the  symbol  i  commonly  designates  electric  current- 
strength,  and  so,  in  order  to  prevent  the  possibility  of  confusion,  the  symbol  j  is 
frequently  substituted  as  the  sign  of  the  imaginary.  Under  such  a  convention,  the 
plane-vector,  or  complex  quantity,  OP\,  would  be  represented  as  i  -\-  j  2.  As,  however, 
in  this  book  we  necessarily  consider  complex  quantities  from  a  broader  viewpoint  than 
that  offered  by  electrical  engineering,  we  shall  use  the  symbol  i  to  denote  the  imaginary 
component,  perpendicularly  rotated  with  respect  to  the  fundamental  X  axis. 

[154] 


EXPLANATORY  TEXT 


Complex  quantities  may  also  be  expressed  in  polar  coordinates,  as  in  Fig.  2,  where 
the  fundamental  reference  axis  OX  is  drawn  in  the  positive  direction  in  the  reference 
plane,  from  the  origin  0,  and  the  circular  angle  5i  is  measured  in  the  positive  or  counter- 
clockwise direction  from  OX  to  OP\.  The  vector  OPi  is  then  specified  in  polar  co- 
ordinates by  its  length  pi  and  by  its  angle  5i.  The  length  p\  is  called  the  modulus  of  the 
vector,  and  the  angle  81  is  called  the  argument.  This  argument  may  be  expressed  in 
circular  radians,  in  degrees-minutes-seconds,  quadrants,  or  any  other  recognized  unit 
of  circular  angle.  Thus,  in  Fig.  2,  the  vector  OPi  may  be  represented  to  polar  co- 
ordinates symbolically  by  pi/8\  or,  using  numbers,  by  2. 236/63°. 26',  where  2.236  is 
the  modulus  to  the  same  scale  of  linear  measure  as  in  Fig.  i,  and  63°. 26'  is  the  argument. 

If  one  and  the  same  complex  quantity  be  expressed  both  in  rectangular  and  polar 

coordinates,  as  follows: 

x  +  iy  =  p/8  (i) 

it  is  evident  that  x  =  p  cos  8,  y  =  p  sin  5,  ;y/a;  =  tan  8,  and  p  =  V  x2  +  y2,  relations  which 
enable  the  coordinates  to  be  changed,  at  will,  from  one  form  to  the  other.  '  Thus  in 
Figs  i  and  2,  xi  =  i,  yi  =  2,  p\  —  VJ  =  2.236,  and  §1  =  tan  -1  (2)  =  63°.26/. 


-X 


FIG.  3.  —  Complex  quantity  —  2  —  » i. 


FIG.  4.  —  Plane-Vector  2.236  e*3-608.  or  2.236^206^34'. 


Similarly,  Figs.  3  and  4  represent  the  complex  quantity  or  plane  vector  OPi  to  rec- 
tangular and  polar  coordinates  respectively.  Here  x%  =  —  2,  yz  =  —  i,  pz  =  Vs  =  2.236 
and  82  =  206°. 34'. 

ADDITION  OF  COMPLEX  QUANTITIES 

One  vector  quantity  is  added  to  another,  by  drawing  it  in  the  reference  plane  from 
the  extremity  of  the  latter  as  origin,  and  then  drawing  a  vector  from  the  origin  to  its 


-X 


-Y 

FIG.  5.  —  Addition  of  two  complex  quantities 
(x  +  »  2)  +  (-  2  -  *  i)  =  -  i  +  »  i. 


FIG.  6.  —  Complex  Addition,  Polar  coordinates. 

Pl/Si  +pJ/5i!  =  p3/53 

2. 236/63  ".26'  +  2.236/2o6°.34'  =  x  .414/135* 
Op  +  pP  =  OP 


free  end.      The  last  named  vector  is  the  required  sum.      Thus,  in  Fig.  5,  the  complex 
quantity  OP2  —  —  2  —  ii    of  Fig    3   is  added  to  the  complex  quantity   OP\  =1+12 


EXPLANATORY  TEXT 


of  Fig  i,  giving  the  resultant  vector  OP  =  —  i  +  ii.  Fig.  6  shows  the  corresponding 
operation  with  polar  coordinate  vectors.  Here  OP2  =  2.  236/206°.  34'  of  Fig.  4  is  added 
to  OPi  =  2.  236/63°.  26'  of  Fig.  2,  to  produce  OP  =  1.414/135°  =  p3/8z  of  Fig.  6. 

On  the  drawing-board,  the  graphic  process  of  adding  vectors  is  as  easily  effected 
when  they  are  expressed  in  polar  as  in  rectangular  coordinates.  But  the  arithmetical 
addition  is  much  more  easily  made  with  rectangular  coordinates.  The  rule  is:  find  the 
vector  sum  by  taking  first  the  sum  of  the  reals,  and  then  the  sum  of  the  imaginaries;  or 


+  xz  +   .    .    .   -f  *„) 


(xi  +  iyj  +  (xz  +  iyz)  +   -    .    •    +  (xn  +  fyB)  = 

+  i  (y*  +  y*  +  .   .   .   +  ?„)  =  2x 
In  the  case  of  Figs.  5  and  6: 

(i  +  «)  +  (-  2  -  «)  =  (+  i  -  2)  +  i(2  -  i)  =  -  i  +  ii  = 
SUBTRACTION  OF  COMPLEX  QUANTITIES 


(2) 


/i3S°. 


Reversing  the  sign  of  a  rectangular  complex  quantity  means  reversing  the  sign  of 
both  its  real  and  imaginary  components.  Reversing  the  sign  of  a  polar  complex  quan- 
tity means  changing  its  argument  by  180°. 

To  subtract  one  complex  quantity  A  from  another  B,  reverse  the  sign  of  A,  and  then 
add  it  thus  reversed  to  B,  by  the  rules  of  addition. 

Y 


o 


FIG.  7.  —  Complex  Subtraction 
(i  +  12)  -  (-  2  -*i)  ..  3  +*3  =OP. 


FIG.  8.  —  Complex  Subtraction,  Polar  Coordinates 

pi /Si  —  p-i/b-i  =  p3/5a 

2.236 763 ".26'  -  2.236/206^34'  =  4.243/45" 

Opt  +  piP  =  OP. 


In  Figs.  7  and  8,  the  vector  Pz  of  Figs.  3  and  4  is  subtracted  from  the  vector  P\  of 
Figs,  i  and  2.  In  Fig.  7,  we  have 

OPi  -  OP2  =  OP. 

(i   +  *2)    -   (-   2    -  *l)    =    (i   +  *2)   +   (2  +  *l). 

=  3  +  *3- 
In  Fig.  8,  pi/3i  —  p2/52  =  psAj 

2.236/63°.  26^  —  2. 236/206°.  34'  =  2. 236/63°.  26'  +  2. 236/26°.  34'. 

=  4.243/45°. 

Here  again  the  process  of  complex  subtraction,  which  is  only  a  slight  modification  of 
complex  addition,  is  very  easily  made  on  the  drawing  board  by  purely  geometric  proc- 
esses, whether  the  quantities  are  rectangular  or  polar.  If,  however,  the  process  is  to 
be  conducted  algebraically,  it  is  much  more  easily  conducted  with  rectangular  coordinates. 

[156] 


EXPLANATORY  TEXT 
MULTIPLICATION  OF  COMPLEX  QUANTITIES 

Two  rectangular  complex  quantities  may  be  multiplied  algebraically  by  the  ordinary 
rules  of  algebra,  remembering  that  i2  =  —  i.     Thus 

(*i  +  fyi)  (*2  +  iyz)  =  (xi  xz  -  yi  y2)  +  i(xi  y2  +  xz  yi).  (3) 

In  Fig.  9,  the  vector  OP\  of  Figs,  i  and  2  is  multiplied  by  the  vector  OP2  of  Figs.  3 
and  4.     The  product  is  the  broken  line  OPz. 

For  (i  +  M)  X  (-  2  -  ii)  =  (-  2  +  2)  -  i(i  +  4)  =  -  *5. 


FIG.  9.  —  Product  and  Quotient  of  Complex  Quantities 

Rectangular  Coordinates 

(i  +.2)  X(-2  -ii)  =  -is  =OP3 

(-2  +11)  -M+i  +  12)  =  -o.8+to.6  =  OP4 


FIG.  10.  —  Product  of  two  Complex  Quantities,  Polar  Coordinates 
2.236  763°. 26'  X  2.236/2o6°.34'  =  5  /270°  =  OPs 


If  the  two  quantities  to  be  multiplied  are  polar;   then 

Pi  /Si  X  PZ/SZ  =  Pi  Pz/8i  + 


(4) 


Or  the  rule  is  form  the  product  of  the  moduli  and  add  the  arguments.     Thus  in  Fig.  10, 

OPi  =  Vj/63°.26'  and  OP2  =  Vs/2o6^3j/  /.  OP3  =  5/27o°.oo/. 


RECIPROCAL  OF  A  COMPLEX  QUANTITY 

The  reciprocal  of  a  rectangular  complex  quantity  can  be  reduced  to  the  standard 
algebraic  form,  by  multiplying  both  numerator  and  denominator  by  the  same  complex 
quantity  with  reversed  imaginary.  Thus: 

i  i  X  (x  -  iy)  _  x  -  iy  _  (      x      \  _  .f      y      \          ,. 

1  yt.2        /  V        /L-2       _J  /t«2       /  \J/ 


*2  + 


x  +  iy       (x  +  iy)  (x  -  iy) 
For  example  if  x  +  iy  =  i  +  12, 

I  _  I          /I    —  J2\    _    !_ 

1+^2  I   +  12  \  I    —   12 )  I 


—12  I  12 

= =    0.2   — 

+  4  5  5 


The  reciprocal  of  a  polar  complex  quantity  is  obtained  by  taking  the  reciprocal  of 
its  modulus,  and  reversing  its  argument.     That  is 


=  -I-  8  = 
p—  p 

Ci57] 


(6) 


EXPLANATORY  TEXT 


For  example: 


V5/63°.26'        VS 


QUOTIENT  OF  COMPLEX  QUANTITIES 

To  find  the  quotient  of  a  complex  quantity  A  divided  by  another  B,  form  the  re- 
ciprocal of  B  and  then  multiply  this  reciprocal  by  A . 
Thus  to  find  (x\  +  iyi)/(x2  +  fy2) 


/  x 


For  example: 


\  __  —  2  —  ii  __  —  2  —  ii  /i  —  i: 
\         i  +  *2  i  -f-  12     \  i  —  i'. 

_   (-   2  -   2)   +  t( 


0  o    ,    •     , 
=   —  o.o  +  i  o.o. 


Thus,  in  Fig.  9, 

The  quotient  of  two  polar  complex  quantities  is  formed  by  taking  the  quotient  of 
their  moduli  and  the  difference  of  their  arguments.     That  is 

P2/02          P2     /E    '          E  /ox 

-=7=  =  —  /fc  ~  Si.  (8) 

Pi/5i      Pi 

Thus  in  Fig.  1 1  we  have  the  quotient  of  OP2  of  Figs.  3  and  4  divided  by  OPi  of  Figs, 
i  and  2,  or 

^  =  i  /I43°.o8/. 

*  A' 


FIG.  n.  —  Quotient  of  two  Complex  Quantities, 

Polar  Coordinates. 
8.836  /2o6°.34>  -5-  2.236  /63°.26'  =  I.o/i43°.o8'  =  O?4 

Pz  ^83 


It  is  thus  evident  that  in  order  to  find  either  the  sums  or  the  differences  of  complex 
quantities,  it  is  desirable  to  have  them  expressed  in  rectangular  coordinates;  while, 
on  the  other  hand,  in  order  to  find  products,  reciprocals,  or  quotients,  it  is  preferable 
to  have  them  expressed  in  polar  coordinates. 


[158] 


EXPLANATORY  TEXT 

POWERS  AND  ROOTS  OF  COMPLEX  QUANTITIES 
It  will  be  evident  from  the  foregoing  that 

(p/8)n  =  pn/n8:  and  \fpjl  =  y~p/8/n  (9) 

operations  that  are  readily  executed  on  polar  complex  quantities. 

CIRCULAR  AND  HYPERBOLIC  FUNCTIONS  GEOMETRICALLY  COMPARED 

Since  the  Tables  in  this  book  are  adapted  for  the  evaluation  of  both  circular  and 
hyperbolic  functions  of  a  complex  variable;  that  is,  either  of  sin  (a;  -f-  iy),  cos  (it  +  iy) 
and  tan  (x  +  iy);  or  of  sinh  (x  +  iy),  cosh  (x  +  iy)  and  tanh  (x  +  iy),  it  may  be 
advisable  to  consider  some  propositions  in  the  comparative  geometry  of  the  circular 
and  hyperbolic  functions,  both  real  and  complex. 

REAL  CIRCULAR  AND  HYPERBOLIC  FUNCTIONS 

The  geometry  of  the  real  circular  functions  sin  x,  cos  x  and  tan  x  relates,  as  is  well 
known,  to  the  motion  of  a  radius  vector  over  a  circle.  The  geometry  of  the  real  hyper- 
bolic functions  sinh  x,  cosh  x  and  tanh  x  relates  to  the  motion  of  a  radius  vector  over  a 
rectangular  hyperbola.  In  Fig.  12,  A  b  c  d  E  g  is  a  circle  x*  +  y2  =  i,  assumed  to  have 
unit  radius,  and  center  0.  As  the  radius  vector  OA  rotates  in  the  positive  or  coun- 
terclockwise direction  about  the  center  0,  it  describes  a  circular  sector  such  as  AOE, 
and  a  circular  angle  /3,  the  tangent  Ef  being  always  perpendicular  to  the  radius 
vector  OE.  The  magnitude  of  the  circular  angle  /3  may  be  defined  in  either  of  two 
ways,  namely:  — 

(1)  By  the  ratio  of  the  circular  arc  length  5  described  during  the  motion,  by  the 
vector's  terminal  E,  to  the  constant  length  p  of  the  radius  vector. 

(2)  By  the  area  of  the  circular  sector  AOE  swept  out  by  the  radius  vector  during 
the  motion. 

According  to  definition  (i),  if  the  radius  vector  generates  any  infinitesimal  angle 
dfi  circular  radians,  by  moving  its  terminal  over  an  infinitesimally  small  circular  arc  ds 

then  <//3  =  —  =  —  circular  radians  (10) 

P        i 

since  the  constant  radius  vector  p  has  been  taken  equal  to  unity  Consequently,  in 
passing  over  any  circular  arc  from  distance  s\  to  distance  s%,  through  a  distance  s%  —  s\  =  sy 
the  total  circular  sector  and  circular  angle  generated  will  of  course  be:  — 


/ds 
—  =  ($2  —  si)  =  s  circular  radians  (n) 


or  the  angle  /3,  as  is  well  known,  becomes  equal  to  the  length  of  the  circular  arc  de- 
scribed, when  expressed  in  circular  radians. 

According  to  definition  (2),  if  in  Fig.  12,  the  radius  vector  of  unit  length  moves  from 
the  initial  position  OA  to  any  position  such  as  OE,  it  will  sweep  out  a  circular  sector  OEA. 

[159] 


EXPLANATORY  TEXT 


If  the  arc  AE1  be  measured  in  the  negative  or  clockwise  direction  equal  in  length  to  the 
arc  AE,  then  it  is  well  known  that  the  area  of  the  double  sector  EOE1  shaded  in  Fig.  12, 
is  equal  to  /3  units  of  area  because  the  area  of  the  whole  circle  is  manifestly  TT  units, 


Consequently,  the  magnitude  of 


2/3 
and  the  shaded  area  is  —  that  of  the  whole  circle. 

27T 

the  angle  (3  expressed  in  circular  radians  is  numerically  twice  the  area  of  the  circular 
sector  AOE  which  it  covers  when  the  circle  has  unit  radius. 


FIG.  12.  —  Circular  Sector  and  Real 
Circular  Functions. 


FIG.  13.  —  Hyperbolic  Sector  and  Real 
Hyperbolic  Functions. 


Turning  now  to  the  hyperbolic  case,  let  A  b  c  d  E  Fig.  13,  be  an  arc  of  a  rectangular 
hyperbola  x2  —  y2  =  i,  assumed  to  have  unit  semi-diameter  OA,  and  center  0.  As 
the  radius  vector  OA  rotates  in  the  positive  or  counterclockwise  direction  with  center  O, 
it  describes  a  hyperbolic  sector  AOE*,  and  also  what  may  conventionally  be  called 
for  convenience  a  "  hyperbolic  angle  "  0.*  The  tangent  Ef  to  the  path  of  the  moving 
terminal  E  always  makes  a  circular  angle  /3  with  the  Y  axis;  or  a  circular  angle  of  2/3 
with  a  perpendicular  to  the  radius  vector.  The  magnitude  of  the  hyperbolic  angle  6 
may  be  denned  in  either  of  two  ways;  namely:  — 

(1)  By  the  ratio  of  the  hyperbolic  arc  length  s  described  during  the  motion,  by  the 
terminal  E,  to  the  integrated  mean  length  of  the  varying  radius  vector. 

(2)  By  the  area  of  the  hyperbolic  sector  AOE  Fig.  13,  swept  out  by  the  radius  vector 
during  the  motion. f 

According  to  definition  (i),  if  the  variable  radius  vector  p  generates  any  infinitesimal 
hyperbolic  angle  dd  by  moving  its  terminal  over  an  infinitesimally  small  hyperbolic  arc 
ds ;  then  | 

J* 

hyperbolic  radians     (12) 


ds 

dd  =  - 
P 


*  It  should  be  pointed  out  that  a  "  hyperbolic  angle  "  in  the  sense  above  denned  is  not  the  opening  be- 
tween two  lines  intersecting  in  a  plane;  but  a  quantity  otherwise  analogous  to  a  circular  angle,  and  the 
argument  x  of  the  functions  sinha;,  cosh  a;,  tanh#,  etc.  The  use  of  the  term  "  hyperbolic  angle  "  can  only 
be  justified  by  its  convenience  of  analogy. 

t  GreenhilFs  "  Differential  and  Integral  Calculus,"  1896,  p.  108. 

I  A  demonstration  of  this  proposition  has  been  given  by  the  author  in  "  The  Application  of  Hyperbolic 
Functions  to  Electrical  Engineering  Problems."  Appendix  L,  p.  250.  University  of  London  Press,  IQII. 

[160] 


EXPLANATORY  TEXT 


Consequently,  in  passing  over  any  hyperbolic  arc  from  distance  Si  to  distance  53  through 
a  distance  s2  —  Si  =  s  ,  the  total  hyperbolic  sector  and  hyperbolic  angle  generated  will  be 


6 


/• 
ds  =  s_ 
p~  pi 


hyperbolic  radians       (13) 


where  p1  is  the  integrated  mean  value  of  p  as  defined  by  the  last  equation.  Any  in- 
finitesimally  small  angle,  whether  circular  or  hyperbolic,  is  therefore  expressed  in  corre- 
ponding  radian  measure  by  one  and  the  same  term  ds/p  ;  but  whereas,  in  the  case  of 
circular  angles,  the  constancy  of  the  radius  vector  makes  the  integral  simply  s/p,  in  the 
case  of  hyperbolic  angles,  the  variation  of  the  radius  vector  makes  the  integral  more 
complex.  Fig.  14  represents  a  circular  angle  of  i  radian  in  five  sections  of  0.2  radian 
each;  while  Fig.  15  represents  a  hyperbolic  angle  of  i  radian  correspondingly  divided. 
The  integrated  mean  radius  vector  of  the  full  sector  A  OF  intersects  the  curve  in  the 
point/,  the  total  length  of  the  arc  A  B  C  D  E  F  being  1.3167  units. 


SINES,  COSINES  AND  TANGENTS  OF  CIRCULAR  AND  HYPERBOLIC  ANGLES 

If,  with  unit  radius,  we  draw  both  a  circular  and  a  rectangular  hyperbolic  sector, 
as  in  Figs.  12  and  13,  and  take  OA  as  the  initial  line  in  each;   then  for  any  position 


FIG.  14.  —  Circular  Sector  of  i  Radian  Subdivided  into  Five 
Sectors  of  0.2  Radian  each. 


FIG.  15.  —  Hyperbolic  Sector  of  i  Radian  Subdivided  into 
Five  Sectors  each  of  o.  2  Radian. 


of  the  radius  vector  such  as  OE,  we  shall  have  in  either  case  the  following  magnitude 
conditions:  — 

The  sine  will  be  equal  to  the  length  of  the  perpendicular  from  the  terminal  of  the 
radius  vector  on  to  the  X  axis. 

The  cosine  will  be  equal  to  the  length  of  the  intercept  on  the  X  axis  made  by  the 
above-mentioned  perpendicular. 

The  versed  sine  will  be  equal  to  the  length  XA,  Fig.  12,  and  A X,  Fig.  13,  between  the 
intercept  on  the  X  axis,  and  the  horizontal  unit  radius. 

[161] 


EXPLANATORY  TEXT 

The  tangent  will  be  equal  to  the  length  of  the  perpendicular  from  the  radius  vector 
(or  radius  vector  produced)  on  to  unit  radius  point  of  the  X  axis.     Thus  in 

Fig.  12,  sin/3  =  XE. 
Fig.  13,  sinh  6  =  XE. 
Fig.  12,  cos  /3  =  OX. 
Fig.  13,  coshfl  =  OX. 
Fig.  12,  tan/3  =  At. 
Fig.  13,  tanh  0  =  At. 

Whereas  the  values  of  sin  /3,  cos  /3  and  tan  /3  fluctuate  periodically  in  sign  as  /3  increases 
from  o  to  a ,  the  values  of  sinh  6,  cosh  6,  and  tanh  6  do  not  change  sign,  the  graphs  of 
the  real  hyperbolic  functions  being  indicated  in  Fig.  16,  as  far  as  6  =  3.0. 


i 

1 

1 

I 

/ 

/ 

/ 

FIG.  16 
Diagram  showing  the  graphs  of  the 
sfne,  cosine,  tangent,  cotangent,  secant 
and  cosecant  of  a  hyperbolic  angle. 

Ordinates'--  Numerical  value  of  the  trigonometrical, 
function 
Abscissas  :-  Numerical  value  of  the  hyperbolic  angle. 

/ 

/ 

1 

/ 

/ 

I 

/ 

/ 

/ 

I 

/ 

1 

/ 

/ 

/ 

/ 

j 

/ 

1 

/ 

/ 

[ 

j 

I 

1 

i 

y 

\ 

Jl 

*  i 

I 

c    \\ 

^- 

'/ 

:    \ 

6^ 

r 

3  

, 

\  " 

V, 

0 

j 

v 
v 

/< 

~) 

0, 

i  - 

-^ 

^ 

v 

"~ 

>.y 

/\ 

v— 

--^NHT 

O.5  - 

"7 

^ 

-4 

^ft 

HI 

'PE 

RB 

OL 

C 

AN 

GLE 

Ljr 

Q      ^ 

/ 

1       1 

••^•w 

'»— 

— 

— 

__^ 

[162] 


EXPLANATORY  TEXT 

BISECTION  or  CIRCULAR  AND  HYPERBOLIC  ANGLES 

If  we  take  any  circular  angle  BOC  Fig.  17,  we  may  of  course  bisect  this  angle  in  either 
of  two  ways:  — 


FIG.  17.  —  Bisection  of  a  circular  sector  in  the  well-known 
manner  by  a  radius  vector  through  the  intersection  of  ter- 
minal tangents,  or  through  the  midpoint  of  the  chord  be- 
tween terminal  points. 


(i)  By  drawing  tangents  bb,  cc,  to  the  curve  at  the  points  B,  C,  respectively,  and 
drawing  the  straight  line  Od  from  the  center  O  through  the  point  of  intersection  d. 


FIG.  18.  —  Bisection  of  a  hyperbolic  sector  by  a  radius  vector  through  the  intersection  of 
terminal  tangents,  or  through  the  midpoint  of  the  chord  between  terminal  points. 

[163] 


EXPLANATORY  TEXT 

(2)  By  drawing  the  chord  BC,  and  marking  the  radius  OD  through  the  midpoint  8 
of  this  chord. 

Similarly,  if  we  take  any  hyperbolic  angle  BDC  Fig.  18,  between  the  points  B  and  C 
of  a  rectangular  hyperbola,  we  may  bisect  this  angle  in  either  of  two  ways:  — 

(1)  By  drawing  tangents  bb,  cc,  to  the  curve  at  the  points  B,  C,  respectively,  and 
drawing  the  straight  line  Od  from  the  center  O  through  the  point  of  intersection  d. 

(2)  By  drawing  the  chord,  BC,  and  marking  the  radius  OD  through  the  midpoint 
5  of  this  chord.* 

COMPARATIVE  GEOMETRY  OF  COMPLEX  CIRCULAR  AND  HYPERBOLIC  FUNCTIONS 

We  have  seen  that  the  real  circular  functions  sin  x,  cos  x,  may  be  derived  from  a 
circle  diagram,  and  that  the  real  hyperbolic  functions  sinh  x,  cosh  x,  may  be  similarly 
derived  from  a  rectangular  hyperbola  diagram.  We  shall  see  that  both  the  complex 
circular  functions  sin  (x  +  iy),  cos  (x  +  iy),  and  the  complex  hyperbolic  functions 
sinh  (x  +  iy),  cosh  (x  -\-  iy),  may  be  derived  from  a  combination  circle  and  hyperbola 
diagram. 

COMPLEX  CIRCULAR  FUNCTIONS 

CONSTRUCTION  FOR  sin  (x  ±  iy),  AND  FOR  sin"1  (u  ±  in) 

In  Fig.  19,  take  OA  =  i  along  the  negative  side  of  the  Y  axis.  From  OA  as  initial 
line,  mark  off  the  circular  angle  x  =  AOB.  From  OB  as  initial  line,  mark  off  the  hyper- 
bolic angle  y  and  its  sector  BOD.  Let  C  be  the  foot  of  the  perpendicular  from  D  on 
OB  produced.  Drop  perpendiculars  from  C  and  D  on  the  axis  of  reals  OX,  at  c  and 
d  respectively.  About  c  as  center,  rotate  cd  positively  through  90°  to  cZ.  Then  will 

*  This  proposition  is  proved  in  GreenhilFs  "  Differential  and  Integral  Calculus,"  Macmillan  &  Co.,  1896, 
page  67,  Fig.  16,  for  the  particular  case  when  the  angle  AOB,  in  our  Fig.  18,  is  zero.  The  demonstration  of 
proposition  (i)  for  the  general  case  of  Fig.  18  is  not  difficult;  but  that  found  by  the  author  is  rather  lengthy. 
The  demonstration  of  the  general  proposition  (2)  is,  however,  brief  and  direct,  as  follows:  — 

Let  0i  be  the  hyperbolic  angle  of  the  sector  AOB. 

Let  02  be  the  hyperbolic  angle  of  the  sector  AOC. 

Then  it  is  required  to  show  that 

S£=hA_  =  hA  = 

Of      OA        i 
But  from  an  inspection  of  the  Figure, 

Be  =  sinh  0i,  Cg  =  sinh  02, 

Oe  =  cosh  0i,  Og  =  cosh  02, 

eB  +  gC      sinh  0i  +  sinh  02 


so  that 

2  2 

Oe  -4-  Og      cosh  0i  +  cosh  02 
and  Of  =  — 

2  2 

_„  jft  _  sinh  0i  -\-  sinh  02 

Of~  cosh  0i  +  cosh  02 

io 
tions,"  1909,  p.  XIV,  Formula  (49). 


(n        I      rt  \ 

which  is  a  known  equivalent  expression  for  tanh  —  ,  see  Becker  and  Van  Orstrand's  "Hyperbolic  Func- 


[164] 


EXPLANATORY  TEXT 

the  complex  vector  OZ  =  Oc  +  icd  be  the  required  circular  sine  of  the  complex  angle 
x  +  iy  radians.  In  the  case  represented,  sin  (i  +  i  i)  =  1.299  +  ^0.635  =  1.446 
/26°.o5.  As  y  varies,  Z  moves  along  the  hyperbola  bZ :  — 

X2  Y* 


sin2  x       cosz  x 
and  as  x  varies,  Z  moves  along  the  ellipse :  — 

X2  Y2 


=  i 


(14) 


cosh2  y    '    sinh2  y 

Both  the  hyperbola  and  the  ellipse  have  as  common  foci  FF',  the  points  X  =  ±  i,  F  =  o. 

7 


FIG.  19.  —  Constructions  for  sin  (*  +  iy)  and  sin-1  (u  =*=  I'D). 


From  the  same  figure,  we  have  also,  if  Oc  =  u  and  cZ  =  iv, 
sin"1  (u  ±  iv)  =  sin"1  Ob  ±  i  cosh"1  OE 

/(IT  uY  +  v2-  Vd  -  «)2  +  v2  \ 

2  / 


=  sn 


±i 


(16) 


^  FZ   ~  F'Z  A     KV          FZ  +  F'Z 

since  Ob  =  and  OE  = 


CONSTRUCTIONS  FOR  cos  (x  +  iy)  AND  FOR  cos"1  (u  +  iv) 

In  Fig.  20,  take  OA  as  unit  distance  along  the  real  or  X  axis,  in  the  positive  direction. 
From  OA  as  initial  line,  describe  the  circular  angle  x,  or  the  circular  sector  AOB  of  area 
x/2.  On  OB  as  initial  line,  describe  the  hyperbolic  angle  y,  or  the  hyperbolic  sector 
area  BOD  of  area  y/2.  Let  C  be  the  foot  of  the  perpendicular  from  D  on  OB  produced. 
Drop  perpendiculars  from  C  and  D  on  the  X  axis  at  c  and  d  respectively.  With  c  as 
center,  rotate  the  line  cd  in  the  positive  direction  through  90°  into  the  position  cZ;  so 
that  ~cZ  =  i.~cd.  Then  the  complex  quantity  OZ  =  Oc  +  *'.cd  will  be  the  required 
circular  cosine  of  the  complex  angle  (x  +  iy)  radians. 

[165] 


EXPLANATORY  TEXT 


In  the  case  represented,  cos  (i  +  i  i)  =  0.834  —  *  0.989  =  i.293\49°.866.      As  y 
varies,  Z  moves  along  the  hyperbola  bZ  defined  by 


X2  Y2 


=  i 


cos^  x       sirr  x 
and  as  x  varies,  Z  moves  along  the  ellipse  ZE,  defined  by 


(18) 


cosh2  y       sinh2  y 
Both  the  hyperbola  and  the  ellipse  have  as  common  foci  FF',  the  points  X  =  ±  i ,  Y  =  o. 


FIG.  20.  —  Constructions  for  cos  (x  =•=  iy)  and  cos"1  («  ±  in). 


From  Fig.  20  we  obtain:  — 

cos"1  OZ  =  cos"1  (u  ±  iv)  =  cos"1  Ob  =F  cosh"1  OE 


E2  1 


(19) 


snce         = 


FZ  ~  F'Z 


and  OE  = 


COMPLEX  HYPERBOLIC  FUNCTIONS 
CONSTRUCTIONS  FOR  sinh  (x  ±  iy)  AND  sinh"1  (u  ±  «'») 

In  Fig.  21,  take  CM  as  unit  length  along  the  real  or  X  axis  in  the  positive  direction. 
From  OA  as  initial  line,  describe  the  circular  angle  y,  or  the  circular  .ector  AOB  of 
area  y/z.  From  OB  as  initial  line  describe  the  hyperbolic  angle  x,  or  the  hyperbolic 
sector  BOD  of  area  x/2.  Let  C  be  the  foot  of  the  perpendicular  from  D  on  OB  pro- 

[166] 


EXPLANATORY  TEXT 

duced.  Drop  perpendiculars  from  C  and  D  on  the  F  axis  at  c  and  d  respectively.  With 
c  as  center,  rotate  the  line  cd  negatively,  or  clockwise,  through  90°  to  cZ.  The 
complex  quantity  OZ  =  Oc  —  i.cd  will  be  the  required  hyperbolic  sine  of  the  complex 
angle  (x  +  iy)  radians. 

In  the  case  represented,  sinh  (i  +  *  i)  =  0.635  +  *  I-2985  =  1.446/63°. 95.     As  x 
varies,  Z  moves  along  the  hyperbola  Zbz:  — 

F2  X2 


=  i 


sin"  y       cos"  y 

and  as  y  varies,  Z  moves  along  the  ellipse  XExy 

F2  Y2 

7  i  -  V 


(20) 


(21) 


cosh2  x       sinh2  # 
The  hyperbola  and  ellipse  are  confocal  at  the  points  F  and/  defined  by  X  =  o,  F  =  ±  i. 


FIG.  21.  —  Constructions  for  sinh  (*  *  iy)  and  sinh"1  (u  =t  in). 

From  Fig.  21  we  also  obtain 

sinh"1  («  ±  w)  =  sinh~l  (cZ  ±  Oc)  =  cosh-1  OE  ±  *  sin"1  Ob. 


=  cosh"1 
db  i  sin"1 


(22) 


since 


[167] 


EXPLANATORY  TEXT 

CONSTRUCTIONS  FOR  cosh  (x  -f-  iy)  AND  cosh"1  (u  +  iv) 

In  Fig.  22,  take  OA  as  unit  distance  along  the  real  or  X  axis  in  the  positive  direction. 
From  OA,  as  initial  line,  describe  the  circular  angle  y,  or  the  circular  sector  AOB  of  area 
y/z.  From  OB,  as  initial  line,  describe  the  hyperbolic  angle  x,  or  the  hyperbolic  sector 
BOD  of  area  x/2.  Let  C  be  the  foot  of  the  perpendicular  from  D  on  OB  produced. 
Drop  perpendiculars  from  C  and  D  on  the  X  axis  at  c  and  d  respectively.  About  c,  as 


FIG.  22.  —  Constructions  for  cosh  (x  =•=  jy)  and  cosh  '  (u  ±  MI). 

center,  rotate  the  line  cd  negatively,  or  clockwise,  through  90°  to  cZ ;  so  that  cZ  =  —  i.cd. 
Then  the  complex  quantity  OZ  =  Oc  —  i.cd  will  be  the  required  cosine  of  the  complex 
angle  (x  +  iy)  radians. 

In  the  case  represented,  cosh  (i  +  i  i)  =  0.834  +  20.989  =  1.293 749°. 866.      As  x 
varies,  Z  moves  along  the  hyperbola  Zbz 


X2 


cos2y       sin2}/ 
As  y  varies,  Z  moves  along  the  ellipse  EZez 

F2 


=  i. 


i 


=  i. 


(23) 


(24) 


cosh2  x       sinh2  x 
The  ellipse  and  hyperbola  are  confocal  at  the  points  A,  a,  denned  by  X  =  ±  i,  Y  =  O. 


Li68] 


EXPLANATORY  TEXT 

From  the  same  figure.     If  Oc  —  u  and  cZ  =  iv 

cosh-1  (u  ±  iv)  =  cosh"1  (Oc  ±  i.cZ)  =  cosh-1  OE  ±  i  cos"1  C 

_i_  ~_\ 

±  *  cos"1 


=  cosh"1         V(l  +  ")2  +  ^  +  V(l  -  U) 

2 


•(25) 


CONSTRUCTIONS  FOR  tan  (x  ±  fy)  AND  tan-1  (u  ±  *») 


In  Fig.  23,  lay  off  along  the  X  axis  a  point  A  distant  tan  x  from  0,  and  also  a  point 
B  such  that  O.5  =  cot  x.  Draw  a  circle  through  A  and  5  having  its  center  on  OX 
at  C.  The  distance  OC  measures  cot  2x  and  the  radius  of  the  circle  is  cosec  2X.  Any 
circle  thus  drawn  will  intersect  the  Y  axis  at  two  points  e  and  /  which  are  at  unit  dis- 


FIG.  23.  —  Constructions  for  tan  (*  ±  iy)  and  tan~'  (u 


tances  from  0.  Then  lay  off  the  Y  axis  two  points  a  and  b,  distant  respectively  tanh  y 
and  coth  y  from  O.  With  center  c  on  the  Y  axis,  draw  a  circle  through  a  and  b.  The 
distance  Oc  will  be  coth  2y,  and  the  radius  of  the  circle  will  be  cosech  2j.  Let  Z  be 
the  point  of  intersection  of  the  two  circles.  Then  OZ  is  the  required  tangent  of  (x  +  iy). 
If  x  is  kept  constant  but  y  is  varied,  the  point  Z  moves  over  the  circle  AZB.  If  on  the 
other  hand  y  is  kept  constant,  but  x  is  varied,  Z  will  move  around  the  circle  aZb  and 
will  make  one  complete  revolution  for  each  increase  of  TT  units  in  x. 

[169] 


EXPLANATORY  TEXT 


In  the  case  represented,  tan  (i  +  i  i)  =  0.2718  +  i  1.084  =  1.118/75°.  916. 

From  Fig.  23  it  is  evident  that  the  angle  AeO  is  equal  to  x,  and  angle  eAO  is  thus 
the  complement  of  x.  Hence  half  the  angle  between  r\_  and  r2  is  the  complement  of  x. 
Moreover  ;y  =  loge  V^/ 


tan-1  (u  ±  iv)  = 


-  tan- 


Therefore,  if  OZ  =  u  + 

1  ( ^  +  tan-1 

\±  v  —  i 


(26) 


CONSTRUCTIONS  FOR  tanh  (x  ±  iy)  AND  tanh-1  (w  ±  n>) 

In  Fig.  24  mark  off  on  the  axis  of  reals  xOX  two  points  T  and  -X"  such  that  the  former 
is  distant  by  tanh  x  and  the  latter  by  coth  x  from  the  origin  O.  Find  the  point  C  mid- 
way between  T  and  X.  Incidentally,  this  point  will  be  distant  coth  2X  from  O.  With 


FIG.  24.  —  Constructions  for  tanh  (x  =*=  »'y)  and  tanh—1  (w  =*=  i») 

center  C  and  radius  CT  =  CX  =  cosech  2x,  draw  the  circle  TXZ.  Mark  off  on  the 
axis  of  imaginaries  yOY,  two  points  t  and  y  such  that  the  former  is  distant  by  tan  y 
and  the  latter  by  cot  y  from  the  origin  O.  Find  the  point  c  midway  between  them. 
Incidentally,  this  point  will  be  distant  cot  2y  from  O.  With  center  c  and  radius  ct  = 
cy  =  cosec  zy,  draw  the  circle  By  At.  This  circle  will  cut  the  axis  of  reals  at  two  points 
A  and  B  distant  each  one  unit  from  O.  It  will  also  intersect  the  circle  TXZ  perpen- 
dicularly at  Z.  Connect  OZ.  This  vector  OZ  is  the  required  hyperbolic  tangent  of 
the  complex  angle  (x  +  iy)  radians. 

[170] 


EXPLANATORY  TEXT 

In  the  case  represented,  tanh  (i  -f-  i  i)  =  1.084  +  20.2718  =  1.118/14°. 084.  As  x 
varies,  Z  moves  along  the  circle  AtB.  As  y  varies,  Z  m^ves  along  the  circle  TZX, 
performing  one  complete  revolution  for  each  TT  units  of  increase  in  y, 

From  the  same  Figure,  if  OZ  —  u  ±  iv  =  op  ±  ipZ,  we  have  tanh"1  (u  ±  iv)  = 
x  ±  iy. 

In  this  case   x  =  loge  Vri/r2 

or   *  =  $log.(ri/ra).  (27) 

and  y  =  -  — -  where  a  is  the  circular  angle  at  Z  between  the  radii  vectores  r,  and  r2. 
2 

Also 


Hence 


DEGREE   OF   PRECISION  OF  TABLES 
INTRODUCTION 

If  a  numerical  quantity,  freed  from  decimals,  is  correctly  expressed  to  within  say 
i  part  in  1000;  i.e.,  i  part  in  io3,  then  this  degree  of  precision  may  conveniently  be 
described  as  precision  of  the  third  order.  In  general,  therefore,  if  a  numerical  quantity 
be  correctly  expressed  to  within  i  part  in  ion,  where  n  is  any  real  positive  number,  its 
precision  is  of  the  nth  order.  The  weekly  statement  of  the  financial  assets  of  a  bank 
might  be  expressed  as  $186,257,361.26  which,  assuming  that  it  is  to  be  taken  as  being 
numerically  correct  to  a  single  cent,  represents  18,625,736,126  cents,  an  apparent 
precision  of  i  in  lo10'27,  or  of  the  10.2  7th  order.  Physical  and  astronomical  precisions 
are  less  ostensibly  pretentious,  however,  and  rarely  exceed  the  6th  order.  Engineering 
computations  are  commonly  satisfied  with  a  precision  of  the  third  order;  although, 
on  rare  occasions,  the  order  required  may  be  the  highest  that  physics  can  attain. 

The  degree  of  precision  corresponding  to  retaining  a  specified  number  of  significant 
digits  correct  within  unity,  in  Tables,  can  only  be  stated  approximately ;  since  it  varies 
with  the  values  of  the  digits.  Thus,  if  we  have  tables  containing  entries  each  of  three 
significant  digits,  correct  to  the  last  digit,  the  lowest  entry  may  be  100  and  the  highest 
999.  The  precision  would  therefore  be  i  in  100  in  the  former  case,  and  practically  i  in 
1000  in  the  latter.  That  is  the  order  of  precision  would  vary  between  the  second  and 
the  third.  The  average  precision  might  be  stated  as  of  the  2. 5th  order.  Such  tables 
of  n  significant  digits  lay  claim  to  an  average  precision  of  the  (n  —  i/2)th  order. 

Many  tables  are,  however,  employed  in  which  the  last  digit  is  stated  to  be  correct  to 
the  nearest  digit;  that  is  within  half  of  unity.  On  that  understanding,  the  precision 
of  say  a  three-digit  table  would  vary  between  1/2  in  100  to  1/2  in  999  or  between  the 
2. 3rd  and  the  3. 3rd  order,  with  a  mean  of  the  2.8th  order.  Consequently,  we  may  say 
that  such  tables,  giving  n  significant  digits,  lay  claim  to  an  average  precision  of  the 
(n  —  o.2)th  order. 

[171] 


EXPLANATORY  TEXT 


DEGREE  OF  PRECISION  PRESENTED  IN  THE  FOLLOWING  TABLES 

The  tables  of  complex  hyperbolic  functions  here  presented  have  been  prvffciied 
with  a  view  to  giving  five  decimal  places  regularly.  This  means  five  significant  digits 
when  the  values  of  the  results  lie  between  o.i  and  unity,  six  significant  digits  when  they 
lie  between  i  and  10,  four  when  they  lie  between  o.i  and  o.oi  and  so  on.  Tables  I  to 
VI  inclusive  were  computed  with  the  aid  of  five-figure  logarithms  of  real  hyperbolic 
functions,  so  that  their  degree  of  precision  is  necessarily  limited  to,  and  must  on  the 
average  fall  below  that  of  such  logarithm  tables,  which,  as  we  have  seen,  is  of  the  4. 8th 


FIG.  25.  —  sinh  (p/5)  expressed  in  polar  coordinates. 

order.      Exclusive  of  such  mistakes  as  may  exist,  they  do  not  claim  a  degree  of  pre- 
cision beyond  the  4. 5th  order. 

Tables  VII  to  XIII  inclusive  were,  however,  computed  for  the  most  part  from  Lig- 
owski's  gudermannian  angles  which  are  tabulated  by  him  for  each  thousandth  of  a 
hyperbolic  radian,  to  the  nearest  hundredth  of  a  second  of  circular  arc.  The  logarithms 
of  the  corresponding  real  hyperbolic  functions  were  then  found  in  the  eight-place 
tables  of  Bauschinger  and  Peters,  which  offer  such  logarithms  for  each  and  every  second 
of  circular  arc.  The  results  were  computed  in  the  formulas  to  at  least  six  significant 

[172] 


EXPLANATORY  TEXT 

digits  and  the  sixth  was  then  frequently  discarded  to  meet  the  needs  9f  the  five-decimal 
table.  Consequently  in  this  group  of  tables,  excluding  such  errors  as  may  exist,  the 
precision  is  on  the  average  of  the  4.8th  order,  and  rises  to  the  5. 8th  order,  when  the 
value  of  the  result  lies  between  i  and  10.  The  average  precision  of  the  second  group 
of  tables  is  thus  about  half  an  order  greater  than  that  of  the  first  group. 

PRECISION  OF  THE  CHARTS  IN  THE  ATLAS 

The  charts  of  the  accompanying  Atlas  have  been  prepared  with  a  view  to  offering 
three  digits  in  the  deduced  quantity,  if  reasonable  care  be  taken  in  their  use.  This 
represents  an  average  degree  of  precision  of  the  2. 5th  order;  or  about  equal  to  that  fur- 


FIG.  26.  —  sinh 


in  polar  coordinates. 


nished  by  an  ordinary  25  cm.  slide  rule.  When  a  higher  degree  of  precision  than  this 
is  needed,  arithmetical  interpolation  in  the  Tables  must  ordinarily  be  resorted  to;  but 
even  then  it  is  desirable  to  obtain  a  preliminary  approximate  value  from  the  Atlas, 
in  order  to  furnish  a  check  against  gross  error. 


[173] 


EXPLANATORY  TEXT 


GRAPHIC  REPRESENTATIONS 

Figs.  25  and  26  present  the  results  obtained  from  Table  I  to  true  polar  coordinates. 
Each  intersection  of  the  curves  corresponds  to  an  entry  in  the  table.  Fig.  25  relates 
to  pages  2,  3,  and  Fig.  26  to  the  rest  of  the  table.  The  curves  of  constant  p  intersect 
those  of  constant  5  perpendicularly.  That  is,  each  intersection  occurs  theoretically 
at  right  angles.  If,  however,  an  attempt  is  made  to  prepare  plates  corresponding  to 
Figs.  25  and  26  on  a  large  scale,  for  a  reasonable  degree  of  precision,  in  rapid  interpola- 
tion by  graphical  inspection,  difficulties  present  themselves.  Firstly,  it  has  been  found 
impracticable  to  procure  polar  coordinate  ruled  sheets  large  enough.  Secondly,  regular 
polar  coordinate  charts  of  the  type  presented  in  Figs.  25  and  26' necessarily  offer  very 
little  graphical  interpolation  precision  at  small  radial  distances  from  the  origin  of  co- 

1     2     3 


^•" 

s* 

i      • 

\ 

\ 

^._ 

1 

Z 

i 

>     4 

FIG.  27.  —  Polar  Coordinate  Diagram  Regular 
Presentation  on  Circular  Sheet. 


FIG.  28.  —  Polar  Coordinate  Dia- 
gram Squared  Presentation  on 
Rectangular  Sheet. 


ordinates,  where  the  radial  lines,  sharply  converging,  crowd  the  diagram.  On  the 
other  hand,  they  offer  relatively  great  apparent  interpolation  precision  at  large  dis- 
tances from  the  origin,  as  the  radii  diverge.  In  preparing  interpolation  charts,  therefore, 
the  author  has  devised  the  scheme  of  using  squared  paper  sheets  for  presenting  such 
polar  coordinate  quantities  as  appear  in  Figs.  25. and  26.  Fig.  27  represents  the  regular 
polar  coordinate  r/_y_  diagram,  in  which  the  lines  of  constant  r  are  circles  concentric 
at  0,  and  the  lines  of  constant  7  are  radii  diverging  from  O.  Fig.  28  represents  the 
corresponding  squared  polar  coordinate  diagram,  in  which  the  lines  of  constant  r  are 
the  vertical  straight  lines  i-i,  2-2,  3-3,  and  4-4.  The  lines  of  constant  7  are  the  parallel 
horizontal  lines,  while  the  origin  point  O  in  Fig.  27  becomes  expanded  into  the  original 
straight  line  o-o  in  Fig.  28.  The  straight,  broken  line  DU  is  transformed  into  the 
curved  line  du,  and,  in  general,  orthogonally  intersecting  curves  of  one  diagram  do  not 
transform  into  orthogonally  intersecting  curves  on  the  other. 

[i74] 


EXPLANATORY  TEXT 

INTERPOLATION  CHARTS 

Plates  IA,  IB,  and  Ic  of  the  Atlas  correspond  to  Table  I  and  present,  to  squared  polar 
coordinates,  the  results  in  that  table.  Each  intersection  of  curves  in  the  plates  corre- 
sponds to  one  entry  in  the  table.  Plate  IA  includes  the  entries  on  pages  2  and  3  of 
the  table;  while  Plates  IB  and  Ic  include  the  entries  in  the  remainder  of  the  table. 
The  curves  of  constant  p  and  constant  5  intersect  one  another  at  various  angles,  but  the 
method  of  interpolation  requires  little  explanation.  The  entering  quantity  will  fall 
within  some  particular  curvilinear  parallelogram.  The  respective  opposite  sides  may 
be  subdivided  into  tenths  in  any  of  the  three  following  ways:  (i)  by  direct  inspectional 
estimate,  (2)  by  graphical  subdivision  on  a  sheet  of  tracing  paper  laid  over  the  chart, 
(3)  by  means  of  a  radiating  decimal  scale  of  lines,  prepared  in  advance,  on  tracing  paper 
or  thin  celluloid.  It  is  not,  in  general,  worth  the  effort  of  attempting  a  closer  subdivi- 
sion than  tenths  of  the  sides  of  any  parallelogram.  The  point  of  intersection  of  lines 
parallel  to  the  sides,  through  the  correct  decimal  points,  is  then  to  be  marked  on  the 
covering  tracing  paper,  or  held  with  a  blunt  pointer,  such  as  a  knitting  needle,  on  the 
chart  itself,  and  the  rectangular  coordinates  of  this  point  read  off  from  the  parallel 
ruling  or  background  of  the  plate.  That  is,  the  charts  are  always  used  with  the  entering 
variable  on  the  curvilinear  coordinates,  and  with  the  result  found  on  the  rectilinear  frame- 
work in  the  background;  except  when  inverse  functions  are  sought,  and  the  procedure 
consequently  reversed. 

TABLE   I 

sinh  (p/8)  =  r£y 
POLAR  HYPERBOLIC  SINES  OF  A  POLAR  VARIABLE 

Table  I,  pages  2  to  7,  gives  the  hyperbolic  sine  of  vectors  up  to  3.0  in  modulus,  by 
steps  of  o.i,  for  each  degree  of  argument  from  45°  to  90°.  The  results  are  also  expressed 
in  polar  coordinates,  as  plane  vectors,  corresponding  to  the  relation:  — 

sinh  (p/5)   =  r/7  (30) 

or  p/d_  =  sinh-1  (7/7).  (31) 

The  graphs  of  the  results  to  true  polar  coordinates  appear  in  Figs.  25  and  26,  where 
the  curves  of  constant  p  always  intersect  orthogonally  the  curves  of  constant  d ;  so  that 
at  any  point  of  intersection  the  angles  of  intersection  are  right  angles.  In  Plates  IA, 
IB,  and  Ic  of  the  Atlas,  the  same  graphs  are  given  to  squared  polar  coordinates,  the 
disadvantages  of  the  distortion  being  more  than  outweighed  by  the  advantages  in 
facility  of  graphic  interpolation.  In  these  charts  the  curves  of  constant  p  do  not  inter- 
sect the  curves  of  constant  8  orthogonally. 


[175] 


EXPLANATORY  TEXT 

INTERPOLATION.     FIRST  CASE.     IN  MODULUS  ONLY 

If  Table  I  is  entered  with  a  vector  quantity  of  more  than  one  decimal  in  modulus 
and  of  some  exact  degree  of  argument,  such  as  2.76/70°;  then  the  result  will  lie  nearly 
on  the  line  between  the  results  for  2.7/70°  and  2.8/70°;  namely,  between  1.2 


and  i.  2136/143°.  005.     A  first  approximation  may  be  obtained  by  proportional  parts 
between  them,  thus  :  — 

Required  sinh  2.76/70° 

by  Table,  sinh  2.80/70°  =  1.2136/143°.  005. 

by  Table,  sinh  2.70/70°  =  1.2031/136°.  489. 

Difference  0.10/70°  =  0.0105      /6°.5i6. 

Proportion  for  0.06/70°  =  0.0063      /3°.Qio. 

sinh  2.70/70°  =  i.  2031/136°.  489. 

Result  sinh  2.76/70°  =  1.2094/140°.  399. 

The  true  value  is  1.2086  /140°.366. 

INTERPOLATION  BY  THE  USE  OF  TAYLOR'S  THEOREM 

When  more  precise  interpolation  is  required  than  that  by  simple  intermediate  pro- 
portion, we  may  use  Taylor's  theorem  in  the  following  form;   since 


-cosh*,  =sinh0,    etc. 


d0 

sinh  (0  +  A0)  =  sinh  0  +  A0  cosh  0  +  sinh  0  +         -  cosh  0  +  .    .  (32) 

2! 
Let  0  =  p/5  and  A0  =  Ap/5. 

Then 
sinh  \(p  +  Ap)/«  j  =  sinh  (p/6)  +Ap/5.  cosh  (p/5)  +~     M  sinh  (p/3)  +  .  .  .    (33) 

The  number  of  correction  terms  to  be  retained  depends  on  the  interval,  and  on  the 
degree  of  precision  desired.  It  is  seldom  that  more  than  two  correction  terms  have  to 
be  retained.  Thus  in  the  case  already  considered:  — 

sinh  (2.  76/70°)  =  sinh  (2.  7/70°)  -f-  0.06/70°.  cosh  (2.  7/70°)  +0.0018/140°.  sinh(2.  7/70°). 
By  Table  II,  page  13,  cosh  (2.7/70°)  =  1.3422/153°.  322.  Consequently  dealing  first 
with  the  first  correction  term  only: 

sinh  (2.76/70°)  =  i.203i/i36°.489  +  0.06/70°  X  1.3422/153°.  322 
=  i.  2031/136°.  489  +  0.08053/223°.  322 
=  i.  2031/136°.  489  (i  +  0.066937/86°.  833) 
=  i.  2031/136°.  489  (i  +  0.003698  +  i  0.06684) 
=  i  2031/136°.  489  (i.  00592  /3°.8io) 
=  i.  2102/140°.  299. 

[176] 


EXPLANATORY  TEXT 

Taking  next  the  second  correction  term  into  account. 

sinh  (2.76/70°)  =  i.  2102/140°.  299  -}-  0.0018/140°  X  i.  203  1/136°.  489 
=  i.  2102/140°.  299  +  0.002166/276°.  489 
=  i.  2102/140°.  299  (i  -+•  0.001789/136°.  190) 
=  i.  2102/140°.  299  (i  —  0.00129  +  i  0.00124) 
=  i.  2102/140°.  299  (0.99871  -f-io.  00124) 
=  i.  2102/140°.  299  (0.99871  /o°.  067) 
=  1.2086/140°.366. 

The  correct  result  is          1.2086/140°.366. 


(34) 


SECOND  AND  GENERAL  CASE.     INTERPOLATION  BOTH  IN  MODULUS  AND  ARGUMENT 

Let  the  entered  quantity  be  sinh  (1.025/80.75°). 

We  have  from  Table  I  the  four  nearest  results  as  follows:  — 

sinh  1.0/80°  =       0.85125/83^.489.             sinh  1.1/80°  =  0.90416/84^.286. 

sinh  1.0/81°  =       0.84940/84°.  156.            sinh  1.1/81°  =  0.90172/84°.  877. 


Difference  for  i°  =  —  0.00185  +  o°.667. 

Proportion  for  0.75°  =  —  0.001388  +  0.500. 

sinh  i.  0/80°.  75  =         0.84986/83°.  989.    sinh  i.i/8o 

sinh  i.i    /8o.75  =         0.90233/84°.  729. 

Difference  for  o.i  =  +   0.05247   /o°.74o. 

Proportion  for  0.025  =  +   0.01312   /o°.i85. 

sinh  i.Q25/8o°.75  =        0.86298/84°.  174. 

The  true  value  =        0.863  72/84°.  166. 


Diff.  for  i°     =  —  0.00244  +  o°-59i- 
—  0.00183  +    0.443. 
=      0.90233 


(35) 


DUAL  INTERPOLATION  BY  THE  USE  or  TAYLOR'S  THEOREM 

Let  the  nearest  tabular  function  be  sinh  0  =  sinh  (p/5_)  and  the  required  function 
sinh  (0  +  A0)  =  sinh  |  (p  +  Ap)/d  +  A8\. 

Then  (p  +  Ap)  /S  +  A3  =  p/S  +  (Ap  +  *PA5).  (36) 

where  the  increment  Ap  +  *pA5  is  taken  with  reference  to  the  vector  axis  p/5.     Refer- 
ring this  increment  to  the  initial  axis  of  reference, 


0  +  A0  =  (p  +  Ap)/5+AS  =  p/5  +  V(Ap2  +  (PA6)2  /&  +  tan'1  -  (37) 


/& 


So  that  A0  =  V  (Ap)2  +  (pAS)2  /8  +  tan"1  (pA3/Ap).  (38) 

When,  however,  A5  is  not  very  small,  the  last  formula  may  contain  an  appreciable 
error,  and  the  following  method  of  deducing  A0,  using  rectangular  complex  quantities, 
is  to  be  preferred. 

[i77] 


EXPLANATORY  TEXT 

Let      0  =  p/_8  =  x  +  iy. 

and        0  +  A0  =  (p  -f  Ap)  /8  +  A3  =  x  +  A*  +  *  (y  +  Ay). 

Then  A0  =  A*  +  i. 


(39) 
(40) 

=    V(Aa)2  +  (Ay)2 /tan-1  (Ay/As).    (41) 

We  then  have  by  Taylor's  theorem,  as  before, 

sinh  (0  +  A0)  =  sinh  0  +  A0.  cosh  0  +  - — -  .  sinh  0  +  - — -  .  cosh  0  +  .  .  .    (42) 

2  3.2 

a  series  in  which  two  correcting  terms  only  need  ordinarily  be  retained.  Thus,  in  the 
example  last  considered,  0  =  1.0/80°  and  0  +  A0  =  1.025/80°. 75.  If  we  form  A0  by 
the  use  of  (37),  we  have  Ap  =  0.025,  A5  =  o°.75  =  0.01309  radian,  pA5  =  0.01309. 


A0  =    V(o-°25)2  +  (o.oi309)2  /8o°  +  tan"1  (0.01309/0.025) 
=  0.02822/80°  +  27°. 637 
=  0.02822 /107°.637. 

If  we  form  A0  by  the  use  of  the  rigid  formula  (41) 
0  +  A0  =  1.025/80°. 75  =  0.164761  +  i  1.0116715. 
0  =  1.0/80°  =  0.173648  +  ^0.9848078. 

A0  =  —  0.008887  +  30.0268637 

=  0.028295/108°  .306. 

Entering  now  the  correction  formula  (42),  we  find  in  the  tables: 

sinh  1.0/80°  =  0.85125/83°. 489,      cosh  1.0/80°  =  0.57991/14°. 521. 

so  that 

sinh  1.025/80°. 75  =  sinh  1.0/80°  +  0.028295     /io8°.3o6  X  cosh  1.0/80° 


6 


=  0.85125/83°. 489  -fr-  0.028295  /io8°.3o6  X  0.57991/14°. 521 
+  0.0004003/2 1 6°. 6 1 2  X  0.85125/83°. 489 
+  o.ooo  ooi  /324°.9i8  X  0.57991/14°. 521. 

It  is  evident  that  in  conformity  with  the  precision  of  the  tables,  only  the  first  two 
correction  terms  need  be  included.      Taking  the  first  into  account,  we  have: 
sinh  i. 025/80°. 75  =  0.85125/83^.489  +  0.028295/108°. 306  X  0.57991/1. 
=  0.85125/83^.489  +  o.oi64o8/i22°.827 
=  0.85125/83/^489  (i  +  o.oi9276/39°.338) 
=  0.85125/83^.489  (i  +  0.014909  +  30.012219) 
=  0.85125/83°. 489  (1.014909  +  30.012219) 
=  0.85125/83^.489  X  i. 01498/0°. 690 
=  0.86400/84°.  1 79. 

[178] 


EXPLANATORY  TEXT 

Taking  up  the  second  correction  term:  — 

sinh  1.025/80°. 75  =  0.86400 784°.  179  +  0.0004003 /2i6°.6i2  X  0.851 25 783°. 489 
=  0.86400/84°. 179  +  0.0003405  /3oo°.  IQI 
=  0.86400/84°. 1 79  (i  +  0.000395/215°. 922) 
=  0.86400/84°. 1 79  (i  —  0.00032  —  20.000232) 
=  0.86400/84°. 1 79  (0.99968  —  to. 000232) 
=  0.86400/84°. 1 79  X  0.99968X0°. 013 
=  0.863727 /84°.166. 
The  true  value  is        0.86372   /84°.166. 

CONCLUSIONS 

In  general,  dual  interpolation  by  simple  proportion,  as  in  (35),  will  give  a  result  of 
the  third  order  of  precision.  In  order  to  secure  precision  of  the  fourth  order,  inter- 
polation by  the  use  of  Taylor's  theorem  as  in  (42)  may  be  required. 

EXTENSION  or  TABLE  BY  USE  or  FORMULA  FOR  28 

Although  Table  I  is  only  carried  as  far  as  3.0  in  modulus  (p  —  3) ;  yet  it  may  be 
used  with  a  little  additional  calculation  in  conjunction  with  Table  II,  for  obtaining 
the  hyp.  sines  of  plane  vector  quantities  of  moduli  up  to  6.0,  by  means  of  the  formula: 

sinh  29  =  2  sinh  0  cosh  6  (43) 

Example:  Required  sinh  5.0/77°,  a  quantity  outside  of  Table  I.  Here  6  =  2.5/77° 
is  within  the  limits  of  the  Table;  so  that 

sinh  5-0/77°  =  2  X  sinh  2.5/77°  X  cosh  2.5/77°  (44) 

=  2  X  0.87843 /i 20°. 891  X  o.96459/i56°.524 
=  2  X  0.87843  X  0.96459/277^.415 
=  1.75686  X  0.96459/277^.415 
=  1. 69465  /277°.41S. 

This  method  ordinarily  calls  for  interpolation  both  in  sinh  6  and  cosh  6.  For  this 
reason,  it  may  be  preferable  to  obtain  the  required  result  by  the  use  of  either  Table 
VII  or  Table  X,  the  limits  of  which  are  less  restricted. 

TABLE  II 

cosh  (p/J>)  =  r[i_ 
POLAR  HYPERBOLIC  COSINES  OF  A  POLAR  VARIABLE 

Table  II  gives  the  value  of  cosh  p/5  between  the  limits  of  p  =  o  and  p  =  3.0  by 
steps  of  o.i,  and  the  limits  5  =  45°  and  5  =  90°,  by  steps  of  i°.  The  graphs  of  these 
quantities,  to  squared  polar  coordinates,  appear  in  Plates  HA  and  HB  of  the  Atlas. 

[179] 


EXPLANATORY  TEXT 

INTERPOLATION  BY  SIMPLE  PROPORTION 

In  general,  as  in  the  case  of  Table  I,  a  very  fair  degree  of  precision  in  interpolation 
can  be  obtained  by  taking  first  simple  proportional  parts  in  argument,  and  then  simple 
proportional  parts  in  modulus. 


Example:  Required  cosh  (0.931 05 1 

We  have  from  Table  II:  - 

cosh  0.9/57°  =      0.88922/23°. 140. 

cosh  0.9/58°  =  0.87602/23°. 003. 
Difference  for  i°  =  —  0.01320  /o°.i37. 
Diff.  for  0.518°  =  -  0.00685  /o°.o7i. 

cosho. 9/57°. 518  =       0.88237 /23°.o69. 


cosh  1.0/57°  = 
cosh  1.0/58°  = 


Difference  for  i°  =  —  0.01626  /o°.o94. 
Diff.  for  0.518°  =  —  0.00844  /o°.Q49. 
cosh  i.o/57°.5i8  =  0.87132 /28°.868. 
cosh  0.9  /57-5i8  =  0.88237  /23.o69. 

Difference  for  o.i  =  —0.01105   /5°-799- 

Diff.   for    0.3105  =  —  0.00343   /i°.8oo. 

cosh  0.93105/57°. 518  =  0.87894 /24°.869. 

The  correct  value  is  0.87837  /24.803. 


INTERPOLATION  OF  TAYLOR'S  THEOREM 

When  a  higher  degree  of  precision  is  required  than  can  be  expected  from  simple 
proportional  parts,  we  may  use  Taylor's  Theorem  in  the  following  form:  — 

cosh  (0  +  A0)  =  cosh  0  +  A0  sinh  0  + 


cosh  0  +  *=f-  sinh  0  +  .  .  .    (45) 
";  3- 

Example:  Required  cosh  0.93105/57°. 518 

having  given  in  Table  II  cosh  0.9/57°  =  0.88922/23°. 140 
and  in  Table  I    sinh  0.9/57°  =  0.85414/64^.^8. 
0.93105/57.518  =  0.500      +  *  0.785398. 
0.900     /57°       =  0.49018  +  20.754804. 

A0  =  0.00982  +  20.030594 

=  0.03214/72°. 196. 

cosh  0.93105/57°. 518  =  cosh 0.9/57°  +  0.03214/72°.  196  X  0.85414/64°. 218 
+  o. 00052 /i44°.392  X  0.88922/23°. 140. 


[180] 


EXPLANATORY  TEXT 

It  is  evident  that  for  the  Tables  here  considered  only  two  correction  terms  need  be 
included.      Taking  up  the  first  correction  term, 

cosh  o.93io5/57°.5i8  =  0.88922 / 2^.140  +  0.03214/22^10)6  X  o.854i4./64°.2i8 
=  0.88922/23°. 140  +  0.02745/136°. 414 
=  0.88922/23°. 140  (i  +  0.03087 /i  13°. 274) 

=   0.88922/23°. 140  (i   —  0.01220  +  *  0.02835) 

=  0.88922/23°. 140  (0.98780  +  i  0.02835) 
=  0.88922/23°. 140  X  0.98780  (i  +  i 0.02870) 
=  0.88922/23°. 140  X  0.98780  X  i. 00041/1°. 645 
=  0.87873 /24°.?85. 

Taking  up  the  second  correction  term:  — 

cosh  0.93105/57°. 518  =  0.87873/24°. 785  +  0.00052/144°. 392  X  0.88922 /23°.i40 
=  0.87873/24°. 785  +  o.ooo46/i67°.532 
=  0.87873/24°. 785  (i  +  o.ooo524/i42°.747 
=  0.87873/24°. 785  (i  —  0.000416  -f-  i 0.000317) 
=  0.87873 /24°.785  (0.999584  +  *  0.0003 1 7) 
=  0.87873  X  0.999584 /24°^7_8£  (i  +  i  0.00032) 
=  0.87837 /24°.785°  X  1/0.018° 
=  0.87837 /24°.8Q3. 

The  correct  value  is      0.87837 /24°.8Q3. 

GRAPHICAL  INTERPOLATION 

For  rapid  but  less  precise  work,  interpolation  may  be  made  by  proportional  parts 
on  Plate  HA  or  Plate  HB,  without  arithmetical  computation. 

TABLE  III 

tanh  (p/3)  =  r/y 
POLAR  HYPERBOLIC  TANGENTS  OF  A  POLAR  VARIABLE 

Table  III  gives  in  polar  coordinates  the  value  of  tanh  p/3  between  the  limits  p  =  o 
and  p  =  3.0  by  steps  of  o.i,  and  the  limits  8  =  45°  and  8  =  90°,  by  steps  of  i°.  The 
graphs  of  these  quantities,  to  squared  polar  coordinates,  appear  in  Plates  HlA  and 
Ills  of  the  Atlas. 

INTERPOLATION  BY  SIMPLE  PROPORTION 

In  general,  as  in  the  cases  of  Tables  I  and  II,  a  very  fair  degree  of  precision  can  be 
obtained  by  taking  first  simple  proportional  parts  in  argument  and  then  simple  propor- 
tional parts  in  modulus. 

Example:   Required  tanh  (0.93105/57°. 518). 

[181] 


EXPLANATORY  TEXT 


We  have  from  Table  III:  — 

tanh  0.9/57°  =  0.96056 /4i°.o78. 

tanh  0.9/58°  =  0.97069/42°. in. 

Difference  for  i°  =  0.01013   /i°.o33. 

Diff.  for   0.518°   =  0.00525   /o°.535. 

tanh  0.9/57°. 518  =  0.96581/41°. 613. 


tanh  1.0/57°  =  1.06648  /37°.Q3S. 

tanh  1.0/58°  =  1.08054  /38°.oo4. 

Difference  for  i°  =  0.01406  /o°.g6g. 

Diff.  for  0.518°  =  0,00728  /o°.5Q2. 

tanh  i.o/57°.5i8  =  1.07376  /37°-537- 

tanh  0.9/57.518  =  0.96581  /4i°.6i3. 


Difference  for  o.i  =  0.10795  /  —  4°-Q76. 
"     for  0.3105  =  O.03352/—  i°.266. 

Inferred  value  of  tanh  0.93105  /57°.5i8  =  0.99933    /40°.347. 
Correct  value  of  tanh  0.93105/57^.518  =     1.0000    /40°.395. 


INTERPOLATION  BY  TAYLOR'S  THEOREM 

For  a  higher  degree  of  interpolation  precision  than  by  simple  proportion,  we  may 
use  Taylor's  theorem  in  the  following  form:  — 

(A0)2 


tanh  (0 


A0)  =  tanh  6  +  A0  sech20  - 
(A0)3 


2! 


2  sech20  tanh0 


2  sech20  (sech20  -  2  tanh20) 


(46) 


Example:  Required  tanh  0.93105/57°.  518. 
having  given  in  Table     I  sinh  0.9/57°  =  0.85414/64.218. 
II  cosh  0.9/57°  =  0.88922/23°.  140. 
Ill  tanh  0.9/57°  =  Q.96o56/4i°.o78. 
Here  A0  =  0.03214/72°.  196,  as  given  by  (41).     Hence  by  Taylor's  theorem  as  far  as 

the  second  correction  term  inclusive, 

0.03214       2.i6 
tanh  0.93105/511518  =  tanh  0.9/52!  +  (0.8892  2) 

(o.o32i4)2/i44°.392 
-  (0.88922)2   /46°.28o 
Taking  up  the  first  correction  term:  — 

tanh  o.93iQ5/57°.5i8  =  0.96056  /4i°.Q78  + 


=  0.96056/41°.  078  +  0.04065/25°.  916 


=  0.96056  /4i°.o78,(i  +  0.04084  —  20.01107) 
=  0.96056/41°. 078  (1.04084  —  i 0.01107) 
=  0.96056  /4I.O78  (1.04090X0°. 609) 
=  0.99985 /4o°.469. 
[182] 


EXPLANATORY  TEXT 

Taking  up  next  the  second  correction  term:  — 

tanh  o.93io5/57°.5i8  =  0.99985/40^469  —  0.00103 /i44°-3Q2  X  0.96056/41^078 

(o.88922)2/46°.28o 

=  0.99985/40^.469  +  0.00103  X  0.96056X40°. 810 

0.79071 

=  0.99985/40°. 469  +  0.00126X40°. 810 

=  0.99985/40°. 469  (i  +  0.00126X81.279) 

=  0.99985/40°. 469  (i  +  0.00019  ~~  *  0.00125) 

=  0.99985/40°. 469  (1.00019  ~  *  0.00125) 

=  0.99985/40°. 469  X  i.oooi9\o°.o72 

=  i.oooo   /4o°.- 

Correct  value  =  1.0000  /40°.395. 

When  more  than  two  correction  terms  have  to  be  retained,  it  is  often  easier  to  deter- 
mine sinh  (8  +  A0)  and  cosh  (6  +  A0)  by  Taylor's  theorem,  as  already  described,  and 
then  to  take  their  ratio  for  tanh  (6  +  A0). 

TABLE   IV 

sinh  0 

POLAR  RATIO  — - —  FOR  POLAR  VALUES  OF  0 
v 

Table  IV  has  been  prepared  by  dividing  the  values  of  sinh  0  found  successively  in 
Table  I  by  their  respective  values  of  0.  The  object  of  the  table  is  to  facilitate  the 
computation  of  the  equivalent  T  or  II  of  any  uniform  alternating-current  line  of  known 
electrical  constants.*  That  is,  the  table  pertains  more  particularly  to  the  applications 
of  hyperbolic  functions  than  to  the  fundamental  properties  of  those  functions.  The 

sinh  (p/5) 

table  gives  the  vector  value  of  -  -  for  the  range  p  =  o  to  p  =  3  by  steps  of  o.i, 

p/5 

and  for  5  =  45°  to  5  =  90°  by  steps  of  i°.  The  graphs  of  the  values  contained  in  the 
tables  are  plotted  to  squared  polar  coordinates  in  Charts  IVA  and  IVs  of  the  Atlas,  for 
rapid  graphic  interpolation. 

INTERPOLATION  BY  SIMPLE  PROPORTION 

A  fair  degree  of  precision  in  interpolation  can  ordinarily  be  obtained  by  first  taking 
simple  proportional  parts  in  argument  and  then  simple  proportional  parts  in  modulus. 

sinh  (1.025/80°. 75) 
Example:  Required  - 


*"The  Application  of  Hyperbolic  Functions  to  Electrical  Engineering  Problems,"  by  A.  E.  Kennelly, 
University  of  London  Press,  1914,  Chap.  III. 

[183] 


EXPLANATORY  TEXT 
We  have  from  Table  IV  the  following  values  of  —  -  —  :  — 

u 

For  1.0/80°  =       0.85125       /3°.48g.  For  1.1/80°  =        0.82196      /4°.286. 

1.0/81°  =       0.84940       /3°.i56.  1.1/81°  =       0.81975       /3°.8y7. 

Difference  for  i°  =  —  0.00185  /—  °°-333-     Difference  for  i°  =  —  0.00221  /—  o°. 

0.75°  =  —  O.OOI39/—  o°.25o.  °°-75  =  ~~  o.  00166  /—  o°. 


For  i.i/8o°.75  =        0.82030       /^°.g 
0.84986      /3°.239.        For  i.o/8o°.75  =        0.84986       /3°.2 


Difference  for  o.i  =  —  0.02956      /o°.74o. 
for  0.025  =  —  0.00739      /o°.i85. 

For  i. 025/80°. 75  =        0.84247       /3°.424. 
Correct  value        =       0.84265      /3.°416. 

When  a  higher  degree  of  precision  is  required  than  can  be  expected  from  proportional 
parts,  the  proper  value  of  sinh  (6  +  A0)  should  be  obtained  by  Taylor's  theorem  as 
already  explained  in  connection  with  Table  I,  and  this  value  divided  by  (6  +  Ad) ; 

because  the  expansion  of  — ,  .       directly,  by  Taylor's  theorem,  does  not  lend  itself 

conveniently  for  computation. 

EXTENSION  FOR  THE  RANGE  OF  THE  TABLE  BY  THE  USE  OF  FORMULA  FOR  26 

Although  Table  IV  is  only  carried  as  far  as  3.0  in  modulus  (p  =  3);  yet  it  may  be 
used  with  a  little  additional  calculation,  in  conjunction  with  Table  II,  for  obtaining 

— — ,  for  vector  values  of  6  with  moduli  up  to  6.0,  by  means  of  the  formula:  — 
a 

sinh  26  =  2  sinh  6.  cosh  6  (47) 

sinh  26       sinh  6 

whence  — - —  =  — - —  .    cosh  6.  (48) 

20  6 

Consequently,  to  find  — - —  for  the  double  of  any  quantity  within  the  range  of  Table 

u 

IV,  find  the  value  of  -       -  for  the  quantity,  by  interpolation  directly  in  Table  IV,  and 

V 

multiply  the  result  by  the  hyperbolic  cosine  of  the  quantity  as  obtained  from  Table  II. 
Corresponding  steps  may  be  taken  with  Charts  II  and  IV. 

sinh  (5.0/77°) 
Example:    Required -; — 5 —  ,  this  being  outside  of  the  limits  of  Table  IV; 

0  *     iii, 

but  not  outside  twice  the  value  therein  obtainable. 
Here  — —  for  0  =  2.5/77°  is  by  Table  IV    0.35137 


e 

and  cosh  6  "      "     "    "        "      II    0.96459 l 

[184] 


Hence 


sinh  (5.0/77°) 
5-0/77° 


EXPLANATORY  TEXT 

=  0-35137 /43°-89i  X  Q.96459/i56°.524. 
=  0.33893  /200°  .415. 


sin  6 


This  procedure  calls  for  interpolation  both  in  -    -  and  in  cosh  6.     For  this  reason 


e 


it  may  be  preferable  to  obtain  the  required  result  by  the  use  of  either  Table  VII  or 
Table  X,  the  limits  of  which  are  less  restricted. 


TABLE   V 

tanh  6 
POLAR  RATIO  — - —  FOR  POLAR  VALUES  OF  0 

u 

Table  V,  like  Table  IV,  has  been  prepared  for  electrical  engineering  applications  of 
hyperbolic  functions,  rather  than  for  developing  these  functions  alone.      It  gives  the 
tanh  (p/6) 


vector  value  of 


p/5 


for  the  range  p  —  o  to  p  =  3.0  in  modulus,  by  steps  of  o.i, 


and  for  the  range  5  =  45°  to  8  =  90°  in  argument,  by  steps  of  i°.  It  was  computed 
directly  from  Table  III  by  dividing  the  resulting  values  successively  by  their  respec- 
tive values  of  6.  The  graphs  of  the  values  in  Table  V  are  presented  to  squared  polar 
coordinates  in  Chart  V,  for  rapid  graphic  interpolation. 

INTERPOLATION  BY  SIMPLE  PROPORTION 

Except  where  a  high  degree  of  precision  in  interpolation  is  required,  it  is  prefer- 
able to  interpolate  first  by  simple  proportion  in  argument,  and  then  by  simple  pro- 
portion in  modulus;  although  this  order  of  operations  may  be  inverted. 

Example:  Required  — -r—  for  6  =  0.93105^ 


9 

We  have  from  Table  V:  — 


For  6  —  0.9/57°  =  1.06729  \i5°.922. 
6  =  0.9/58°  =  1.07854  \i5°.889. 


For  6  =  1.0/57°  =  i-  06648  \i^< 


Difference  for  i°  =  o.oii25\—  0.033. 


for  0.518°  =  o.oo583\—  0.017. 


For  0  =  0.9/57°. 518  =  J-°7312 


-9°5- 


6  =  1.0/58°  =  i.o8o54\i9°.996. 

Difference  for  i°  =  0.01406  \o°.o3i. 

"     for  o°.5i8  =  0.00728  \o°.oi6. 

For  0  =  i.o/57°.5i8  =  1.07376X19°.  981. 

6  =  o.9/57°-5i8  =  1.07312  \i5-905. 

Difference  for  o.i  =  0.00064   \4°-o76. 
"  for  0.03105  =  0.00020  \i°.266. 


For0  =  o.93iQ5/57°-5i8  =  1.07332X17^171. 
Correct  value, 
[185]  • 


1. 07406  \17°.123. 


EXPLANATORY  TEXT 

When  a  higher  degree  of  precision  is  needed  than  simple  proportion  can  give,  it  is 
preferable  to  find  the  proper  interpolated  value  for  tanh  Q  from  preceding  tables  and 

then  to  divide  by  0;   since  the  function  -        -.  —  —  r  —  does  not  lend  itself  to  expansion 

(0  +  A0) 

by  Taylor's  theorem  in  a  simple  form. 

Tables  IV  and  V  jointly,  with  their  respective  graphs  in  the  Atlas,  enable  the  equiva- 
lent T  or  II  of  any  uniform  alternating-current  line  in  the  steady  state,  at  a  single  fre- 
quency, to  be  completely  determined,  provided  0  does  not  exceed  six  radians  in  modulus 
(5  lying  between  45°  and  90°)  ;  because  although  in  both  tables,  0  is  not  carried  beyond 

three  radians;  yet  —  —  —  can  be  found  by  extension  up  to  six  radians,  and  in  the  formu- 
0 

las  for  deducing  the  equivalent  T  or  II,  —  -  —  -  has  only  to  be  carried  to  half  the  modulus 

0 

,  sinh  0 
of-0-' 

The  following  example  may  illustrate  the  use  of  Tables  IV  and  V  either  with  or 
without  the  aid  of  the  graphic  interpolation  Charts  IV  and  V  of  the  Atlas.  An  alter- 
nating-current line  of  uniform  electrical  constants  is  250  km.  long  and  has,  at  a  certain 
frequency,  a  total  conductor  impedance  of  565.711  784°.  777  ohms,  associated  with  a 
total  distributed  insulation  admittance  of  4.3707  X  io~3/9o°  mhos.  Its  hyperbolic 
angle  is  therefore  Vs.  65711  X  4.3707  X  io~1/i74  .777  =  1.5724  787°.  388  hyperbolic 
radians.  Interpolating  either  from  the  tables  or  the  Charts  IV  and  V,  we  obtain 

sinh0  .tanh  ()       tanh  0.7862/87^88 

=  0.638  / 


.  _  0.786a/8 

If  we  multiply  the  conductor  impedance  by  —  -  —  ,  we  have 

u 

565.  711  784°.  777  X  0.638/2°.  6  =  360.69/87°.  377    ohms,   and  if    we   multiply  half  the 

insulation  admittance  by  —  ff).  g   ,  we  have 

Iff/ 

2.1854  X  io-3/9o°  X  i.27\i°.5  =  2.78  X  io-3/88°.g  mhos  =  359-77\88°.5  ohms. 

If  now  we  apply  an  artificial  condenser  leak  of  2.78  X  io-y88°.5  mhos  to  each  end  of  a 
localised  impedance  coil  of  3  60.  6  9/87^.3^7  ohms,  we  obtain  the  "equivalent  II"  of  the 
line  at  the  frequency  considered,  and  such  a  combination  of  localised  impedance  and  ad- 
mittances would  behave  exactly  like  the  line,  at  its  terminals,  or  outside  them,  so  as  to 
be  capable  of  replacing  the  line  in  any  electrical  system,  at  that  frequency. 


[186] 


EXPLANATORY  TEXT 

TABLE   VI 
POLAR  FUNCTIONS  OF  POLAR  SEMI-IMAGINARY  QUANTITIES 

A  semi-imaginary  quantity  is  a  complex  numerical  quantity  which,  when  expressed 
in  rectangular  coordinates,  has  equal  real  and  imaginary  components;  or,  when  ex- 
pressed in  polar  coordinates,  has  an  argument  of  45°.  That  is  x/±  45°  =  a  rt  i  a. 
The  interest  of  the  table  pertains  primarily  to  the  application  of  hyperbolic  functions 
to  uniform  alternating-current  lines  of  negligibly  small  linear  inductance  and  leakance, 
a  case  approximated  to  by  cabled  lines  at  low  frequencies.  The  table  was  first  pub- 
lished by  the  author  in  the  transactions  of  the  International  Electrical  Congress  of 
St.  Louis  (1904).  The  arguments  of  the  results  are  given  in  degrees  and  minutes,  and 
not  in  degrees  and  decimals  like  the  rest  of  the  tables. 

The  table  gives  the  hyperbolic  sine,  cosine,  tangent,  cosecant,  secant,  and  cotangent 
of  the  vector  x/4$°  for  the  range  x  =  o  to  x  '=  20.5,  by  steps  in  x  of  o.i  up  to  x  =  6, 
and  of  0.05  beyond  that  point.  At  x  =  6,  the  values  of  the  hyp.  sine  and  cosine  so 
nearly  coincide,  that  they  are  taken  as  equal  in  the  table,  thus  bringing  sech  x  and 
cosech  x  into  equality  as  well  as  tanh  x  =  coth  x  =  i.  Graphs  of  the  functions  are 
given  in  Chart  VI  as  far  as  x  =  4,  approximately. 

INTERPOLATION  BY  SIMPLE  PROPORTION 

In  general,  interpolation  may  be  quickly  effected  by  simple  proportional  parts  of 
modulus  since  the  argument  is  constant  at  45°.  This  procedure  is  sufficiently  evident 
to  require  no  exemplification. 

INTERPOLATION  OF  TAYLOR'S  THEOREM 

When   precise   interpolation   is   necessary,    we   have   the   following   expansions  for 

f  (0  +  A0) 


sinh     (*  +  A*)  /4S!      =  sinh  (2/45°)  +  (Ax)  /4S°.   cosh  (*/45°)  + 


sinh  (*/45°)  +  -=r/ilS°.   cosh  (*/45°)  +  .....  (49) 

O  " 

cosh  |  (*  +  A*)/45°}    =  cosh   (x/4S°  +  (A*)/45_°.    sinh(*/45°)  +  ^fl^L< 

(Ar^3 

cosh  (*/45°)  +  ^fr/ias!-  sinh  (*^0)  +  —  (5o) 

,  tanh     (*  +  A*)/4,S!      =  tanh 


2  sech2  (»/4S°).  tanh  (s/4Q  -  Zl^.0.  2.  sech2  (*/45°). 

J  sech2  (*/45°)  -  2  tanh2  (s/4S°)  *+....  (51) 

Example:  Required  cosh  (3.1/45°),  having  given  in  Table  VI 
sinh  (3.0/45°)  =  4.1986  /i  2o°.48/. 
cosh  (3.  0/45°)  =  4-U43  /i22°.i6/. 
[187] 


EXPLANATORY  TEXT 

Here  cosh  (3.1/45!)  =  4.i443/i22°.i6'  +  0.1/45°  X  4-1986 /i2o°.48' 

0.01/90°                                         o.ooi  , 
+  -  -  X  4.i443/i22°.i6'  H —  /i35°  X  4.1986 / 120°. 48' 

=   4.1443 /I22°.l6'  +  4.1986/120°. 48   (0.1/45°  +  0.00017/135°) 
+  4. 1443  /I  22°. 1 6'  (0.005/90°) 

=  4. 1443 /i 2 2°. 1 6'  (i  +  20.005)  +  4.i986/i65°.48'  (o.i  +  20.00017) 

=  4.i443/i22°.i6'  ( i. 0000/0°. 1 7')  +  4. 1986/165°. 48'  (o.i  /o°.oi') 
=  4.1443  /i  22^33'  +  o.4i99/i65°.49/ 

:2°.33/  (i  +  o.  10132/43°.  16') 


=  4.1443  /i  22°.33/  (i  +  0.07378  +  *  0.06944) 
=  4.1443  /I22°.33/  (1.07378  +  *  0.06944) 
=  4.I443/I22°.33'  X  1.0760  /3°.42; 


which  is  in  substantial  agreement  with  the  tabulated  value  of  cosh  3.1. 

Beyond  x  =  6,  the  value  of  either  sinh  (3/45°)  or  cosh  (#/45!)  was  computed  from 
the  formula:  — 


sinh  (s/45°)  =  cosh  (s/45°)  =  J  —  /T/T  radians.  (52) 

v  ^ 

where  e  =  2.71828  .  .  . 


X  6  141.14 

Thus,  with  x  =  7,  —  T=  =  4.9498,  -  =  -  =  70.57  at  the  argument  of  4.9498  cir- 

\j  2  ^2 

cular  radians  =  283°.  36';  so  that:  — 

sinh  (7/45°)  =  cosh  (7/4.5!)  =  7Q.57/283°.36/ 
which  coincides  with  the  tabulated  value  in  Table  VI. 

TABLE   VII 

sinh  (x  +  iy)  =  u  +  vo 
RECTANGULAR  HYPERBOLIC  SINES  OF  A  RECTANGULAR  VARIABLE 

Tables  I  to  VI  contain  certain  restrictions  in  range  which  limit  their  general  appli- 
cation. They  are  primarily  designed  to  cover  particular  applications  of  hyperbolic 
functions  to  electrical  engineering.  Tables  VII  to  XIV,  however,  are  free  from  such 
restrictions,  and  are  intended  to  furnish  the  circular  as  well  as  the  hyperbolic  sine,  cosine, 
and  tangent  of  a  complex  angle,  and  to  furnish  this  result  either  in  the  rectangular  or 
polar  form.  That  is,  they  furnish:  — 

sinh  (x  +  iy)  \   .      ,     .  .        .      .      .     ,  , 

;          .  ;  >  in  the  f  orm  u  +  vo  ;  also  in  the  form  r  /j 
or  sin    (x  +  iy)  j 

cosh  (x  -\-  iy)\   .    ^    f  •       ,      •    ,1     e  / 

.  (  }  in  the  form  u  +  vo\  also  in  the  form  r  /y 
or  cos    (x  +  iy)  J 

tanh  (x  +  iy)  \    .  .      ,     ,  , 

.  (  >  in  the  form  u  +  tv:  also  in  the  form  r  /y 
or  tan    (x  +  ly)  } 

[188] 


EXPLANATORY  TEXT 

between  the  limits,  for  the  hyperbolic  functions,  x  =  o  and  x  =  ±  10,  by  steps  of  0.05 
and  between  the  limits  y  =  o  and  y  =  ±  oc  ,  by  steps  of  0.07854  =  TT/^O. 

PERIODIC  PROPERTIES  OF  THE  RECTANGULAR  COMPLEX  HYPERBOLIC 

SINES  AND  COSINES 

It  is  well  known  that  sinh  \x  +  i  (y  -f  2mr)\  =  sinh  \x  +  iy\  (53) 

and  cosh  \x  +  i  (y  +  2mr)  \  =  cosh  \x  +  iy\  (54) 

where  n  is  any  integer. 

This  means  that,  keeping  x  constant,  the  values  of  the  hyp.  sine  and  hyp.  cosine 
repeat  themselves  as  iy  passes  through  increments  of  i.2ir\  or  they  are  periodic  functions 
of  iy,  having  the  period  2ir  i. 

The  matter  may  be  visualised  more  clearly  from  geometrical  reasoning.  Consider- 
ing the  exponential  form  of  the  hyperbolic  cosine, 

c  *  +  'V   _L   e  -<*  +  '») 

cosh  (x  +  iy)  =-        — —    .  (55) 


e*       .         e~1 
This  may  written  in  the  form:  — .  e  iy  -\ e~iv.        If  x  be  kept  constant,   we 

require  to  study  the  changes  produced  in  this  form  of  the  hyp.  cosine  by  varying  y. 

q' 
$ 

%sZ»**>* 

-a        d^~  -A 

• 
P 

FIG.  29.  —  Geometrical  constructions  for  cosh  (x  +  iy)  and  sinh  (*  +  iy). 

In  Fig.  29,  OA  is  an  initial  line  and  OP  a  radius  vector  of  length  or  modulus  €*/2> 
multiplied  by  e  '»;  that  is  rotated  positively  about  O,  from  OA  through  a  circular  angle 
of  y  radians.  Similarly,  op  is  a  vector  of  length  or  modulus  €~*/2  rotated  negatively 
through  a  circular  angle  of  y  radians  from  the  initial  line  Oa.  The  equation  (55)  states 
that  the  hyperbolic  cosine  is  the  plane  vector  sum  of  OP  and  op,  or  O'p'  in  the  Figure. 
If  now  we  steadily  increase  the  value  of  y,  leaving  x  constant,  we  cause  OP  to  rotate 
steadily  counterclockwise,  and  also  op  to  rotate  steadily  clockwise,  through  Ay  cir- 
cular radians.  When  Ay  =  2ir,  both  OP  and  op  will  have  made  one  complete  revo- 
lution and  will  have  returned  to  their  initial  positions  indicated.  Consequently,  the 
value  of  cosh  \x  +  i  (y  +  271-)  \  repeats  that  of  cosh  \x  +  iy \. 

€  x  +  iy  _   f  -(x  +  iy) 

Since  sinh  (x  +  iy)  =  -  (56) 

the  same  reasoning  applies;   but  the  vector  op  is  added  in  the  negative  or  reversed  di- 
rection;  so  that  O'q'  is  the  hyperbolic  sine  of  x  +  iy. 

The  above  mentioned  periodic  property  of  the  hyp.  sine  and  cosine  has  been  utilized 
for  shortening  the  tables  of  those  functions  by  reducing  the  circular  angle  y  of  Fig.  29 

[189] 


EXPLANATORY  TEXT 

from  radians  to  quadrants.  That  is,  any  complex  angle  x  +  iy  represented  by  a  point 
P,  and  radius  vector  OP,  in  the  complex  plane  XY,  Fig.  30,  is  first  transferred  to  a 
new  complex  plane  XQ,  Fig.  31,  at  the  point  p  =  x,  q,  by  keeping  x  the  same  in  both 

planes,  but  making  the  points  -,  —  ,—,—,...  etc.,  on  the  Y  axis  of  the  XY  diagram, 

2222 

become  the  points  i,  2,  3,  4  ...  etc.,  on  the  Q  axis  of  the  XQ  diagram.       Thus  if 

x  +  iy  =  2.5  +  *  6. 2832, 
x  +  iq  =  2.5  +  i  4.00 

where  4.00  is  underscored  to  indicate  quadrant  measure,  instead  of  the  ordinary  radian 
measure. 

"Y 


FIG.  .30. 


FIG  3r. 


Transference  of  a  Complex  Quantity  from  the  XY  to  the  XQ  Plane. 

In  the  case  indicated  by  Fig.  30,  x  +  iy  =  3  +  i  9    and  x  +  iq  —  3  +  *  5.74  in  Fig.  31. 

Consequently,  after  a  complex  angle  has  been  transferred  from  the  complex  plane 
XY  to  the  complex  plane  XQ,  the  values  of  either  sinh  (x  +  iq)  or  cosh  (x  +  iq)  exactly 
repeat  themselves  for  each  4  units  of  increase  in  q;  or  with  reference  to  Fig.  31  for  each 
4  quadrants  of  increase  in  the  circular  angle  instead  of  6.2832  .  .  .  radians.  The 
operation  of  transferring  the  complex  angle  from  the  XY  to  the  XQ  plane  may  there- 
fore be  described  as  quadranting  y;  i.e.,  changing  the  expression  of  y  from  circular 
radian  units  to  circular  quadrant  units. 

All  of  the  Tables  VII  to  XIII  inclusive  require  to  be  entered  in  terms  of  x  +  iq; 
so  that  the  complex  entering  value  has  to  be  quadranted  by  dividing  its  imaginary  or 

y-component  by  the  numeric  v/2  =  1.57079 This  preliminary  step  occupies 

a  certain  extra  time  and  effort;  but  it  actually  economises  the  total  time  and  effort 
involved.  If  the  tables  were  computed  for  x  -f-  iy,  they  would  have  to  be  repeated  in 
bulk  for  each  TT  radians,  or  2  quadrants,  increase  in  y.  In  electrical  engineering  appli- 

[  190] 


EXPLANATORY  TEXT 

cations,  y  frequently  rises  to  100  radians,  and  might  easily  be  much  greater  than  100. 
In  order  to  go  up  to  100  radians,  the  bulk  of  the  Tables  VII  to  XIII  would  have  to  be 
increased  about  thirty  fold.  Altogether,  aside  from  the  greatly  increased  bulk  and  expense 
of  such  tables,  the  extra  time  and  effort  consumed  in  turning  over  the  numerous  pages 
would  be  comparable  with  that  saved  by  eliminating  the  preliminary  step  of  quadranting 
the  imaginary  component  or  dividing  it  by  ir/2. 

RULES  FOR  THE  USE  OF  TABLE  VII 

Express  the  "  angle  "  whose  hyperbolic  sine  is  required  in  the  form  of  an  ordinary 
rectangular  complex  quantity  x  +  iy. 

Quadrant  the  imaginary  component  y  through  the  process  of  dividing  it  by  ir/2; 
i.e.,  transfer  the  quantity  from  the  XY  to  XQ  plane;  so  that  the  new  expression  of  the 
complex  quantity  is  x  +  iq;  where  q  =  y/i. 57079.  .  .  . 

If  q  is  greater  than  4.6,  divide  by  4  and  retain«only  the  remainder.  If  the  remainder 
exceeds  2,  subtract  2  therefrom,  and  apply  a  negative  sign  to  the  result  found  in  the 
table.  A  change  of  2  quadrants  simply  reverses  the  sign  of  the  result.  If  the  remainder 
on  the  other  hand  does  not  exceed  2,  enter  Table  VII  with  it,  and  take  out  the  result 
with  unchanged  sign. 

•  Example:  Required  the  hyperbolic  sine  of  0.65  +  i  25.75.  Here  x  =  0.65  and 
y  =  25.75.  That  is  y  is  25.75  circular  radians.  Reduce  this  to  quadrants  through 
dividing  by  1.57079.  .  .  . 

25-75  *  log  25.75     =  1.4107772. 

1.57079    ...      log          TT/2    =   0.1961199. 

log  16.393  =  1.2146573. 
The  quadranted  yalue  x  +  iq  =  0.65  +  i  16.393 

NOTE.  —  It  is  found  convenient  to  underscore  quadrantal  quantities  to  distinguish 
them  from  radianal  quantities. 

Rejecting  quadrant  multiples  of  4,  i.e.,  16  in  this  case,  we  enter  Table  VII  with 
x  -|-  iq  =  0.65  +  i  0.393.  The  nearest  entry  to  this  is  x  =  0.65,  q  =  0.4,  for  which 
the  hyperbolic  sine  is  0.56368  +  *  0.71639,  an  ordinary  rectangular  complex  quantity 
on  the  U  V  plane.  Interpolation  should  be  made  in  this  result  to  meet  the  change 
from  q  =  0.40  to  q  =  0.393,  as  wiH  De  explained  later. 

Second  Example:  Required  sinh  (x  +  iy)  =  sinh   (1.15  +  i  10.10). 

.    /               .  10.10  \ 
Quadranting  the  imaginary,  sinh  (x  +  iq)  =  sinh  (  1.15  +  * =  1 

=  sinh    (1.15  +  i  6.430). 

Rejecting  4'$  from  the  imaginary  =  sinh    (1.15  +  i  2.430). 
Deducting  2  from  the  residual  and  chang- 
ing the  sign =  —  sinh  (1.15  -f  *'o-43°)- 

*  This  operation  would  ordinarily  be  effected  with  the  slide-rule,  when  a  high  degree  of  precision  is  not 
aimed  at. 

[191] 


EXPLANATORY  TEXT 

We  now  enter  Table  VII  with  x  =  1.15  and  q  =  0.43.  The  nearest  entry  is  x  =  1.15 
q  =  0.45,  the  result  for  which  is  1.08037  +  *  1.12836.  But  we  must  apply  a  negative 
sign  to  the  whole  of  this  result  because  of  the  2  rejected  in  the  quadrantal  residuum. 
Hence, 

sinh  (1.15  +  i  10.  10)  =  —  (1.08037  +  *  1-12836) 

=  —  1.08037  —  i  1.12836  =  u  +  i  v 

except  for  the  interpolation  from  q  =  0.45  to  q  =  0.43.      The  operation  of  interpolation 
will  be  discussed  later  on. 

Third  Example:    Required  sinh  (x  +  iy)  =  sinh    (3.60  +  i  18.1). 

Quadranting  the  imaginary,  sinh  (x  +  iq)  =  sinh  (  3.60  +  i—    '—^  } 
*  \  i-57°<V 

=  sinh    (3.60  +  i  11.523). 

Rejecting  4'$  from  the  quadrants  =  sinh    (3.60  +  ^'3.523). 
Deducting  2  from  the  residual  imaginary 

and  changing  the  sign  ..............  =  —  sinh  (3.60  +  i  1.523). 

Entering  Table  VII  with  x  =  3.6  and  q  =  1.523,  the  nearest  entry  is  x  =  3.6  and 
q  =  1.5,  for  which  the  result  is  —  12.92978  +  *  12.94910.  But  applying  the  negative 
sign  to  this  result  because  of  2  deducted  from  the  quadrantal  imaginary,  and  we  have 

finally:  — 

sinh  (3.60  -+-  i  18.1)  =  —  (—  12.92978  +  i  12.94910) 

=  12.92978  —  i  12.94910  =  u  +  i  v 

except  for  the  interpolated  correction  from  q  =  1.500  to  q  =  1.523,  to  be  considered 
later. 

RANGE  OF  THE  TABLE 

Table  VII  extends  by  steps  of  0.05  in  x  up  to  x  =  3.95,  and  in  Table  XIII  up  to 
x  =  4.0.  In  y,  the  range  is  indefinitely  great;  because  after  dividing  y  by  ir/2  so  as 
to  reduce  it  to  quadrant  measure,  all  multiples  of  4  are  rejected.  From  o  to  2,  in  the 
remainder,  the  table  gives  the  result  directly  and  from  2  to  4,  by  change  of  sign  in  the 
total.  Cases  of  x  greater  than  4.0  are  dealt  with  in  connection  with  Table  XIV. 

REPETITIONS  IN  THE  TABLE 


If  sinh  {  x  +  i-  —  a        =       u  +  iv  (57) 

I  \2          /  ) 

it  is  easy  to  show  that: 

sinh  I  *  -M  (  -  +  0  )  \  =  —  u  +  iv.  (58) 

I  \2          /  J 

It  follows  that  in  any  column  of  Table  VII,  the  entry  for  q  =  (i  —  a)  is  the  same  as 
that  for  q  =  i  +  a  except  for  a  change  in  the  sign  of  u.  Consequently,  the  table  might 
have  been  reduced  to  half  its  present  size,  if  the  responsibility  for  making  this  change  of 
sign  had  been  left  to  the  reader.  It  was  considered,  however,  that  since  the  reader  is  al- 
ready charged  with  the  duty  of  applying  a  negative  sign  to  the  total  result  when  the  q 

[  192] 


EXPLANATORY  TEXT 

residuum  lies  between  2  and  4,  the  retention  of  the  full  size  of  the  present  table  was 
warranted,  especially  as  the  duplication  of  the  text  in  each  column  provides  a  certain 
check  upon  the  numerical  work  of  tabulation. 


INTERPOLATION  BY  SIMPLE  PROPORTION 

As  a  first  approximation,  interpolation  may  be  effected  by  simple  proportion,  first 
in  regard  to  x  and  second  in  regard  to  q. 

Example:   Required  sinh  (0.15  -f  3*0.25),  having  given: 

sinh  (0.2  -f  3*0.2)  =  0.19148  +  3*0.31522.     sinh  (0.2  +  i 0.3)  =  0.17939  +  3*0.46310. 
sinh  (o.i  +  3*0.2)  =  0.09526  +  3*0.31056.     sinh  (o.i  +  3*0.3)  =  0.08925  +  3*0.45626. 

Diff.  for  o.i  x  =  0.09622  +  3*0.00466.  Diff.  for  o.ix  =  0.09014  -f-  3*0.00684. 

Diff.  for  0.052;  =  0.04811  +  3*0.00233.  Diff.  for  0.052;  =  0.04507  +  3*0.00342. 

sinh  (0.15  +  i 0.3)  =  0.13432  +  3*0.45968. 

sinh  (0.15  +  3*0.2)  =  0.14337  H~  *  0.31289.  sinh  (0.15  +  3*0.2)  =  0.14337  -f-  3*0.31289. 

Diff.  for  q  o.io  =  —  0.00905  +  3*0.14679. 

Diff.  for  q  0.05  =  —  0.00453  +  i  0.07340. 

sinh  (0.15  +  3*0.25)  =  0.13885  +  3*0.38629. 

Correct  value  =  0.13910  +  3*0.38700. 

INTERPOLATION  BY  TAYLOR'S  THEOREM 

When  a  higher  degree  of  precision  is  desired  than  that  which  can  be  expected  by 
simple  proportion,  we  may  use  Taylor's  theorem  in  the  following  form:  — 


sinh  (0  +  A0)  =  sinh  0  +  A0  cosh  0  +  ^-  sinh  0  +  z=f-  cosh  0  +   .  .  .    (59) 
sinh  {  (x  +  iy)  +  (Ax  +  i  Ay)  \  =  sinh  (x  +  iy)  -f-  (Ax  +  i  Ay)  cosh  (x  +  iy) 


.    (Ax  +  3*  Ay)2    .  ,    f         ...    (Ax  +  iAy)z       ,    ,         .  ,  ,,.% 

+  -     —  —  —    -  sinh  (x  +  3y)  +  -      —.  —    -  cosh  (x  +  ly)  +    •  •  •         (60) 

*  O  * 

Quadranting  imaginaries  on  both  sides;  or  transferring  to  the  XQ  plane, 
sinh  |  (x  +  iq)  +  (Ax  +  i  Aq)  \  =  sinh  (x  +  iq)  +  \  Ax  -f  i  (IT/  2)  Aq)  \  cosh  (x  +  iq) 


!A*  +  3*(7r/2)A;y!2    .  \Ax  +  3  Or/2)  Aq\3         ,    ,  .,  , 

+  L  -  smh  (x  +  iq)  +  L  A-t-         *-  cosh  (x  +  iq)  +  .  .  .     (61) 

=  sinh  (x  +  iq)  +  A'0  cosh  (x  +  iq)  +  -  —  ~-  sinh  (x  +  iq) 

+  ^  cosh  (*  +  iq)  +  .  .  . 
3! 

where  A'0  =  Ax  +  iAy  =  Ax  +  i  (IT/  2)  Aq.  (62) 

[193] 


EXPLANATORY  TEXT 
Example  (i):  With  A^  =  o. 

Required  sinh  (0.15  +  i  0.2),  having  given  in  Table  VII  and  in  Table  VIII: 
sinh  (o.i    +  i  0.2)  =  0.09526  +  i  0.3 1056. 
cosh(o.i     +  3*0.2)  =  0.95582  +  i 0.03095.     Then  by  (60); 

sinh  (0.15  +  *  0.2)  =  sinh  (o.i  +  i  0.2)  +  0.05  cosh  (o.i  +  i  0.2)  -\ — 

sinh  (o.i  +  3*0.2)  -\ —      — -  cosh  (o.i  +  3*0.2) 
o 

=  0.09526  +  3*0.31056  +  0.05  (0.95582  +  30.03095) 

+  O.OOI25   (0.09526  +  3*0.31056)   +  O.OOO02   (0.95582   + 

3*  0.03095) 

=    I.OOI25   (0.09526  +  3*0.31056)  +  0.05002  (0.95582  +  3*0.03095) 
=    (0.09538  +  30.31095)   +   (0.04781   +  30.00155) 

=  0.14319  +  3*  0.3 1250 
which  is  the  correct  tabular  value  of  sinh  (0.15  +  i  0.2)  in  Table  VII. 

Example  (2):  With  Ax  =  o. 

Required  sinh  (o.i  +  3*0.25),  having  given  in  Table  VII  and  in  Table  VIII: 
sinh  (o.i  +  3*0.2)     =  0.09526  +  3*0.31056. 
cosh  (o.i  +  3*0.2)     =  0.95582  +  3*0.03095.     Then  by  (62); 
sinh  (o.i  +  3*0.25)  =  sinh  (o.i  +  3*0.2)  +  3*0.05  X  1.5708  X  cosh  (o.i  +  3*0.2) 

.,  (0.05  X  1.5708)2    .       f  . 

+  32-  —  sinh  (i.o  +  30.2) 

2 1 

.    .,  (0.05  X  1.5708^       ,   t  . 

+  33-^ — - —  -  cosh  (i.o  +  30.2) 

3! 

=  0.09526  +  3*0.31056  +  3*  X  0.07854  (0.95582  +  3*0.03095) 

0.00617  /  <:\ 

(0.09526  +  0.31056) 


2 
.  O.OO048 


(0.95582  +  30.03095) 


6 
=  (0.09526  +  30.31056)  (i  —  0.00309) 

+  3*  (0.95582  +  3*0.03095)  (0.07854  —  0.00006) 

=  0.99691  (0.09526  +  3*0.31056)  +  0.07848  (—  0.03095  +  3*0.95582) 
=  0.09497  +  3*0.30960  —  O.OO243  +  3'O.075OI 
=  0.09254  +  3*0.38461. 

The  tabular  value  is     0.09254  +  i  0.38460. 

Example  (3) :  Interpolation  for  both  Ax  and  A^. 
Required  sinh  (0.15  +  i  0.25),  having  given 

sinh  (o.i  +  3*0.2)  =  0.09526  +  3*0.31056  by  Table  VII 
and  cosh  (o.i  +  3*0.2)  =  0.95582  +  3*0.03095  by  Table  VIII. 

Here  A6  in  formula  (59)  =  (0.05  +  i  0.05) 

and  A'0  in  formula  (62)  =  (0.05  +  i(ir/2)  X  0.05)  =  (0.05  +  30.07854). 


EXPLANATORY  TEXT 

Thus:  - 

sinh  (0.15  +  ^0.25)  =  sinh  (o.i  +  10.2)  +  A'0  cosh  (o.i  +  io.z 

(A'0)  2 
+        ,    •  sinh  (o.i  +  io.2)  +  .  .  . 

A'0  =  0.05  +  i  0.07854. 
(A'8)2  =  +  0.0025  —  0.00617  +  i  0.00785 
=  —  0.00367  +  io.  00785. 

(A'0)2 

=  —  0.00184  ~t~  ^ 


(A'8)3  =  (0.05  +  i  0.07854)3  =  —  0.00080  +  *'o.oooio. 
(A'8)* 

—  7  —   =    —  0.00013  +  *  0.00002. 

f  (A'0)2  "> 

sinh  (0.15  -f  10.25)  =  sinh  (o.i  +  i  0.2)  j  i  -f-  -  —  p  +  •  •  .  > 

+  cosh  (o.i  +  *  0.2)  j  A'0  +  ^^  +  .  .  .  | 

\  3  / 

=  (0.09526  +  i  0.31056)  (0.99816  +  i  0.00393)  =  0.09386  +  i  0.31036 
+  (0.95582  +  i  0.03095)  (0.04987  +  i  0.07856)  =  0.04524  +  i  0.07664 

=  0.13910  +  i  0.38700. 
The  correct  tabulated  value  is   =  0.13910  +  i  0.38700. 

EFFECTS  OF  CHANGES  OF  SIGN  IN  THE  ENTERING  QUANTITY 
Table  VII  expresses  the  relation 

sinh  (x  +  iq)  =  u  +  iv.  (63) 

(a)  If  x  be  taken  with  negative  sign,  we  have 

sinh  (—  x  +  iq)  =  —  u  +  iv  (64) 

so  that  changing  the  sign  of  the  real  component  entering  the  table  changes  the  sign  of 
the  real  component  in  the  result;  but  leaves  the  sign  of  the  imaginary  component 
unchanged. 

(b)  If  q  be  taken  with  negative  sign,  we  have 

sinh  (x  —  iq)  =  u  —  iv  (65) 

so  that  changing  the  sign  of  the  imaginary  component  in  the  entering  quantity  changes 
the  sign  of  the  imaginary  component  in  the  result,  leaving  the  sign  of  the  real  com- 
ponent unchanged. 

• 

(c)  If  both  x  and  q  be  taken  with  negative  sign,  we  have 

sinh  (—  x  —  iq)  =  sinh  \  —  (x  +  iq)  \  =  —  u  —  iv  =  —  (u  +  «0         (66) 

so  that  changing  the  sign  of  the  total  entering  quantity  changes  the  sign  of  the  total 
result. 

The  facts  may  be  summed  up  by  saying  that  changes  in  the  sign  of  the  entering 
quantity  produce  corresponding  changes  of  sign  in  the  result. 

[195] 


EXPLANATORY  TEXT 

CIRCULAR  SINES  OF  COMPLEX  "  ANGLES  " 

Since,  as  is  well  known:  — 

sin  6  =  —  i  sinh  (iff)  (67) 
we  have 

sin  (x  +  iy)  =  —  i  sinh  (ix  —  y)  (68) 

=        *  sinh  (y  —  ix).  (69) 

Consequently,  in  order  to  find  the  circular  sine  of  the  complex  quantity  (x  +  iy),  enter 
Table  VII  for  sinh  (y  +  ix),  which  on  being  quadranted,  becomes  sinh  \  y  +  ix/(ir/2)  \ 
and  let  the  result  be  (u  +  iv).  Then  sinh  (y  —  ix)  =  u  —  iv  and  sin  (x  +  iy)  =  v  +  iu. 
In  other  words,  invert  the  entering  components,  and  then  invert  the  components  of  the 
result. 

Example:  Required  sin  (i  +  i  2)  from  Table  VII.     Here  6  =  (i  +  i  2). 

Enter  the  Table  with  —  sinh  (id)  =  sinh  (—  id)  =  sinh  (2  —  i  i). 

Quadranting  the  imaginary,  we  enter  the  table  with  (x  —  iq)  =  (2  —  i  0.6366). 
The  nearest  entry  is  (2  —  i  0.65),  for  which  the  hyp.  sine  is  given  as  1.89503  —  i  3.20780. 
Consequently,  sin  (i  +  i  2)  =  3.2078  -f-  i  1.89503,  except  in  so  far  as  interpolation  is 
needed  to  reduce  sinh  (2  —  i  0.6366)  from  sinh  (2  —  3' 0.65).  In  this  way  any  circular 
sine  of  a  complex  quantity  can  always  be  obtained  from  the  table  of  hyperbolic  sines, 
between  the  limits  of  o  and  ±  4  in  y,  and  of  o  and  ±  °o  in  x. 

GRAPHIC  INTERPOLATION  BY  MEANS  OF  CHARTS  VIlA,  Vila,  VIIc 

Charts  VII-VIII  A,  B,  and  c,  serve  for  the  evaluation  of  either  sinh  (x  +  iq)  or 
cosh  (x  +  iq),  according  to  the  axis  of  reference  selected.  Thus,  taking  Chart  VII- 
VIIIs,  if  this  is  held  with  the.  line  SS  as  the  axis  of  reference  or  initial  line;  then  by 
comparison  with  the  entries  in  Table  VII,  it  will  be  found  that  sinh  (x  +  iq)  can  be 
read  from  it  directly  over  the  range  q  =  o  to  q  =  4,  beyond  which  the  values  repeat 
themselves  indefinitely.  On  the  other  hand,  if  the  chart  be  turned  through  90°,  so 
as  to  bring  the  line  CC  as  the  axis  of  reference,  it  will  be  found  by  comparison  with  the 
entries  of  Table  VIII,  that  cosh  (x  +  iq)  can  be  read  from  it  directly  over  the  range 
q  =  o  to  q  =  4. 

Chart  VII-VIIlA  gives  sinh  (x  +  iq)  and  cosh  (x  +  iq)  for  values  of  x  up  to  about 
0.9.  Chart  VII-VIIIs  gives  the  corresponding  results  for  values  of  x  up  to  about 
x  =  2.  Finally,  Chart  VII-VIIIc  provides  for  values  of  x  up  to  x  =  4.  In  all  of 
these  charts,  interpolation  can  be  made  for  both  x  and  q  to  o.oi,  by  direct  inspection. 
The  graphs  on  these  charts  are  undistorted,  since  they  give  complex  functions  as  results, 
in  rectangular  coordinates.  The  curves  therefore  always  intersect  orthogonally,  and 
they  represent  a  confocal  system  of  ellipses  and  hyperbolas,  the  common  foci  being  at 
two  points  at  unit  distances  from  the  center,  along  one  of  the  reference  axes.  The  curvi- 
linear rectangles  into  which  the  charts  are  divided  have  pairs  of  sides  the  ratio  of 
whose  lengths  tends  to  the  value  ir/2. 

If  the  preliminary  process  of  quadranting  the  imaginary  of  the  entering  quantity 
were  not  adopted;  that  is,  if  the  graphs  were  entered  in  terms  of  (x  +  iy),  instead  of 

[196] 


EXPLANATORY  TEXT 

(*  +  iq) ;  then  it  would  be  necessary*  to  have  a  new  chart  for  each  range  of  2ir  units  in 
3>;  or  some  16  sets  of  Charts  A,  B,  and  C,  in  order  to  reach  y  =  100.  That  is,  48  charts 
would  have  to  be  computed,  prepared,  drawn,  lithographed,  bound,  sold  and  operated 
instead  of  the  3  charts  actually  presented.  Moreover,  if  y  were  needed  greater  than 
100,  the  set  of  48  would  fail;  whereas,  working  with  quadrant  imaginaries,  the  three 
charts  serve  up  to  indefinitely  great  values  of  q  and  y, 

GRAPHIC  CHART  VII-VIIlA 

This  chart  corresponds  to  Tables  VII  and  VIII  at  least  as  far  as  x  =  0.9,  or  for 
pages  42  to  45,  and  58  to  61  of  this  book.  To  find  hyperbolic  sines  from  the  chart, 
place  it  facing  the  observer  with  the  axis  OO  vertical.  This  is  the  major  axis  of  all  the 
ellipses  shown.  Starting  from  this  central  axis  towards  the  right  hand,  the  successive 
ellipses  marked  o.i,  0.2,  0.3,  etc.,  represent  values  of  x;  while  the  successively  rising 
hyperbolas  o.i,  0.2,  represent  values  of  q.  These  values  of  q  will  be  found  to  extend 
over  two  quadrants.  Enter  the  chart  on  the  curvilinear  coordinates  for  x  and  q.  At 
the  proper  intersection  read  off  the  u  and  v  coordinates  of  the  rectilinear  ruling,  u  being 
the  abscissas  and  v  the  ordinates. 

Conversely,  to  find  sinh"1  (u  +  iv)  within  the  limits  u  =  o  and  «  =  ±i,»=oto 
v  =  2.0,  enter  the  chart  with  the  same  aspect  on  the  rectilinearly  ruled  coordinates 
and  read  off  at  the  proper  intersection  the  curvilinear  values  taking  x  on  the  ellipses 
and  q  on  the  hyperbolas. 

To  find  hyperbolic  cosines  from  the  chart,  rotate  it  clockwise  90°;  so  as  to  have 
the  axis  OO  horizontal.  Then  enter  on  the  curvilinear  coordinates  with  x  on  the  ellipses 
and  q  on  the  hyperbolas.  The  first  and  fourth  quadrants  only  will  be  presented  to  the 
observer;  but  from  the  symmetry  of  the  diagram,  it  will  be  easy  to  reverse  the  chart, 
so  as  to  present  the  second  and  third  quadrants.  Read  off  the  result  on  the  rectilinear 
background  using  u  for  abscissas  and  v  for  ordinates. 

Conversely,  to  find  cosh"1  (u  +  iv)  from  the  chart  with  the  axis  OO  horizontal,  enter 
on  the  rectilinear  background  and  read  off  at  the  proper  intersection  from  the  curvilinear 
coordinates  in  x  and  q,  taking  the  ellipses  as  parts  of  the  it-system  and  the  hyperbolas 
as  part  of  the  ^-system. 

GRAPHIC  CHART  VII-VIIIs 

This  chart  gives  the  graph  of  the  functions  sinh  (x  +  iq)  and  cosh  (x  +  iq)  from 
x  =  0.8  at  least  as  far  as  x  =  2.05  along  the  ellipses  and  from  q  =  o  to  q  =  °o  by  virtue 
of  successive  rotations.  In  this  and  the  following  charts,  the  numerical  values  of  q  are 
all  underscored,  an  indication  which  may  serve  readily  to  distinguish  the  imaginaries 
q,  from  the  reals,  x. 

*  A  single  set  of  charts  entered  in  terms  of  (x  +  iy)  could  be  used  up  to  y  —  6.2832  in  one  revolution, 
and  could  be  used  for  all  larger  values  by  throwing  out  multiples  of  2*-.  This  operation  of  dividing  y  by  2T 
would,  however,  take  as  much  time  as  the  operation  of  quadranting,  and  would  also  lead  to  a  dissymmetrical 
chart  in  the  hyperbolas. 


EXPLANATORY  TEXT 

In  all  of  the  Charts  VII  to  IX  inclusive,  the  curvilinear  rectangles  all  tend  to  have 
sides  in  the  ratio  TT  :  2;  that  is  the  long  side  approximates  to  being  1.57  times  the  short 
side.  In  IXA  exceptions  are  found;  because  extra  curvilinear  coordinates  are  supplied. 

To  find  sinh  (x  +  iq)  from  Chart  VII-VIIlB,  hold  the  minor  axis  55  horizontal. 
Enter  on  the  curvilinear  coordinates  with  x  on  ellipses  and  g  on  hyperbolas.  At  the 
proper  intersection  read  off  on  the  rectilinear  background  in  u  and  v.  Proceed  inversely 
to  find  the  inverse  function  sinh"1  (u  +  iv). 

To  find  cosh  (x  +  iq)  from  the  same  chart,  hold  the  major  axis  CC  horizontal.  Enter 
on  the  curvilinear  coordinates  with  x  on  ellipses  and  q  on  hyperbolas.  Read  off  on  the 
rectilinear  background. 

All  four  quadrants  appear  in  this  and  the  following  chart,  so  that  it  is  not  necessary 
to  limit  the  value  of  q  to  less  than  2  quadrants. 

GRAPHIC  CHART  VII-VIIIc 

This  Chart  gives  the  graph  of  sinh  (x  +  iq)  and  cosh  (x  +  iq)  from  x  =  2.0,  at  least 
as  far  as  x  =  3.90.  The  procedure  is  precisely  the  same  as  that  for  VII-VIIlB  already 
described. 

TABLE   VIII 
cosh  (x  +  iq)  =  «  +  iv 

RECTANGULAR  HYPERBOLIC  COSINES  or  A  RECTANGULAR  VARIABLE 
Table  VIII  may  be  regarded  as  an  inversion  of  Table  VII;   because: 

cosh  6  =  —  i  sinh  (6  +  iir/z)  (70) 

or  in  quadrant  imaginaries, 

cosh  6  =  —  i  sinh  (6  +  i  i).  (71) 

That  is  the  hyp.  cosine  of  any  complex  quantity  (x  +  iq)  is  —  i  times  the  hyp.  sine 
of  that  quantity  with  an  additional  quadrant  in  the  imaginary.     Thus 
cosh  (0.5  +  io,6)  =  —  isinh  (0.5  +  i  1.6) 

=  —  *  (—  0.42158  +  20.66280) 
=  +  0.66280  +  *  0.42158. 

All  of  the  entries  in  Table  VII  thus  reproduce  themselves  by  inversion  in  corresponding 
parts  of  Table  VIII,  a  fact  which  serves  as  a  numerical  check  upon  both. 

In  order  to  find  the  value  of  cosh  (x  +  iy),  quadrant  the  imaginary  quantity  y,  by 
dividing  it  with  IT/ 2,  as  in  entering  Table  VII.  The  complex  quantity  (x  +  iy)  will 
now  be  expressed  as  (x  -\-  iq);  or  will  in  effect  have  been  transferred  from  the  XY  to 
the  XQ  plane.  Next  throw  out  multiples  of  4  from  q,  so  as  to  leave  a  remainder  less 
than  4.  If  this  remainder  exceeds  2,  deduct  2  from  it,  but  change  the  sign  of  the  total 
result  thereupon  deduced.  •  If  the  remainder,  however,  is  not  greater  than  2,  then, 
the  result  is  taken  directly  from  the  table. 

Example:  To  find  cosh  (1+25        )  =  cosh  (x  +  iy). 
Quadranting,  we  have  cosh  (i  +  13.183)  =  cosh  (x  +  iq). 
Deducting  2  from  q,  cosh  (i  +  i  1.183). 

[198] 


EXPLANATORY  TEXT 

With  this  we  enter  Table  VIII.  The  nearest  entry  is  x  +  iq  =  i  +^1^2,  the  result 
for  which  is  —  0.47684  +  i  1.11768.  This  has  to  be  corrected  by  interpolation  from 
q  =  1.2  to  q  =  1.183.  Reverse  the  sign  of  the  result  to  0.47684  —  i  1.11768  for  the 
deduction  of  2  quadrants. 

Example  2:   Required    cosh  (0.25  +  2*30        )  =  cosh  (x  +  iy). 
Quadranting,  this  becomes  cosh  (0.25  +  i  19.099)  =  cosh  (x  +  iq). 

Rejecting  imaginary  quadruples  =  cosh  (0.25  +  2*3.099) 

=  cosh  (x  +  iq). 

Deducting  2  quadrants  =  cosh  (0.25  +  i  1.099) 
=  —  cosh  (x  +  iq). 

The  nearest  entry  is  0.25  +  ii.i  for  which  the  result  is  (—  0.16135  +  *  0.24950). 
Applying  the  negative  sign  on  account  of  the  two  deducted  quadrants,  the  final  result 
is,  neglecting  interpolation, 

cosh  (0.25  +  i  30)  =  0.16135  ~~  *  o-2495o  =  u  —  iv. 

INTERPOLATION  BY  SIMPLE  PROPORTION 

A  first  approximation  can  be  obtained  by  interpolating  according  to  simple  pro- 
portion. 

Example:   Required  cosh  (0.55  +  2*0.55)  =  cosh  (x  +  iq) 
having  given 

cosh  (0.6  +  20.5)  =  0.83825  +  i  0.45018.  cosh  (0.6  +  i  0.6)  =  0.69680  +  10.51506. 
cosh  (0.5  +  2*0.5)  =  0.79735  +  i  0.36847.  cosh  (0.5  +  20.6)  =  0.66280  +  2*0.42158. 

Diff.  for  x  o.i     =  0.04090  +  20.08171.  Diff.  for  x  o.i     =  0.03400  -f-  20.09348. 

Diff.  for  x  0.05  =  0.02045  ~l~  20.04086.  Diff.  for  x  0.05  =  0.01700  +  2*0.04674. 

cosh  (0.55  +  i  0.6)  =  0.67980  +  20.46832. 
cosh  (o.55  +  2o.jj)  =  0.81780  +  20.40933.        cosh  (0.55  +  2*0.5)  =  0.81780  +  10.40933. 

Diff.  for  q  cxi     =  —0.13800  +  2*0.05899. 
Diff.  for  (70.05  =  —0.06900  +  2*0.02950. 

cosh  (0.55  +  2*0.55)  =      0.74880  +  20.43883. 
Correct  value  =      0.75018  +  *  0.43963. 

INTERPOLATION  BY  TAYLOR'S  THEOREM 

For  a  higher  degree  of  precision  than  simple  proportion  affords,  reference  may  be 
had  to  Taylor's  theorem  in  the  following  form  :  — 


cosh  (6  +  A0)  =  cosh  6  +  A0  sinh  Q  +         -  cosh  6  +  sinh  0  +  .  .  . 

2!  3! 

cosh  {  (x  -f-  iy)  +•  (Ax  +  2'  Ay)  \  =  cosh  (x  +  iy)  +  (Ax  +  iAy)  sinh  (x  +  iy) 

+  (^±iM!  cosh  (,  +  ^  +  (a»-M*y)'  sinh  (,  +  iy)+...     (7ia) 


EXPLANATORY  TEXT 

Quadranting  imaginaries  on  both  sides,  or  transferring  to  the  XQ  plane, 

cosh  \  (x  +  iq)  +  (A*  +  iAq)  |  =  cosh  (x  +  iq)  +  (Ax  +  iAq  ir/2)  sinh  (x  +  iq) 

(Ax  +  iAq  7T/2)2 
+  -  *         - 


u  t          •  \ 
cosh  (*  +  ^q) 


,    (Ax  +  3*A<7  ir/2)3   .  ,    ,         .  . 
+  -        f*-       -  sinh  (x  +  39)  +  . 

O  " 

=  cosh  (x  +  iq)  -f  A'0  sinh  (x  +  iq) 
cosh  (x  +  iq)  +  .  .  . 


where  A'0  =  (Ax  +  *  Aj)  =  (Ax  +  *  A^  x/a).  (7ib) 

Example:  Required  cosh  (0.5  +  3*0.55)  =  cosh  (x  +  iq) 

having  given  cosh  (0.5  +  ^0.5)  =  0.79735  +  i  0.36847  in  Table  VIII 
and  sinh  (0.5  +  ^0.5)  =  0.36847  +    0.79735  in  Table  VII. 
Here  Ax  =  o,    Aq  =  10.05,    A'0  =  30.05  X  1.5708  =  30.07854. 

cosh  (0.5  +  io^)  =  cosh  (0.5  +  io.£     i  +  +  +  .  . 


+  sinh  (0.5  +  3'o^    A'0 
A'0     =       3*0.07854. 


(A'0)2 
(A'0)2  =     —  0.00617.  ~~T~  =     ~  0-00309 


2! 

(A'0)3 
(A'0)3  =  —  3*0.00048.  — p  =  —  3*0.00008. 

(A'0)4 

(A'0)4  =     +  0.00004.  — i      =          o.ooooo. 

4! 

cosh  (0.5  +  3*0.55)  =  (o.79735  +  ^  0.36847)   (i  -  0.00309) 

+  (0.36847  +  ^' 0.79735)   (30.07854  —  3*0.00008) 
=  (°-79735  +  *  0.36847)  0.99691 

+  (0.36847  +  i  0.79735)  3*0.07846 
=  0.79489  -f-  3*0.36733  +  3*0.02891  —  0.06256 
=  0.73233  +  3*0.39624. 
The  tabulated  value  =  0.73233  +  «  0.39624. 

In  view  of  the  similarity  of  the  interpolation  operations  by  Taylor's  theorem  to 
those  already  discussed  in  relation  to  Table  VII,  further  examples  are  probably  not 
needed. 

EFFECTS  OF  CHANGES  OF  SIGN  IN  THE  ENTERING  QUANTITY 

Since  if          cosh  (      x  +  iy)  =  u  +  iv  (72) 

cosh  (—  x  +  iy}  =  u  —  iv  (73) 

cosh  (      x  —  iy)  =  u  —  iv  (74) 

cosh  (—  x  —  iy)  =  u  +  iv  (75) 

changing  the  'sign  of  either  the  real  or  imaginary  entering  component  only  changes  the 
sign  of  the  imaginary  component  in  the  result;  while  changing  the  sign  of  the  enter- 
ing quantity  as  a  whole,  has  no  effect  on  the  sign  of  the  result. 

[  200] 


EXPLANATORY  TEXT 

CIRCULAR  COSINES  OF  COMPLEX  "  ANGLES  " 

It  is  well  known  that  if  6  be  any  angle,  real  or  complex, 

cos  6  =  cosh  (i6).  (76) 

Consequently, 

cos  (x  +  iy)  =  cosh  (—  y  -f  ix)  (77) 

or,  quadranting  the  imaginary  component, 

cos  (x  +  iy)  =  cosh  (—  y  -f-  i  2x/ir)  =  u  +  iv.  (78) 

To  find  the  circular  cosine  of  any  complex  quantity  x  +  iy,  we  enter  Table  VIII 
with  (—  y  -f  *  x/ 1. 5 708).  The  result  is  the  desired  cosine. 

Example:   Required  cos  (0.4  +  i  1.2). 

Thus  we  require  cosh  (—  1.2  +  i  0.2546). 

We  now  enter  Table  VIII  with  x  =  —  1.2  and  q  =  0.2546  the  nearest  entry  being 
x  =  —  1.2  and  q  =  0.25,  for  which  the  result  is  1.67283  —  *  0.57765. 

Hence  cos  (0.4  +  *  1.2)  =  1.67283  —  i  0.57765  neglecting  interpolation  from  q  =  0.25 
to  q  =  0.2546. 

GRAPHIC  CHART  INTERPOLATIONS 

The  use  of  the  Graphic  Charts  VII-VIIlA,  B,  c,  for  hyperbolic  cosines  has  already 
been  described  in  connection  with  sines,  on  pages  197-198. 

TABLE   IX 

tanh  (x  +  iq)  =  u  +  iv 

RECTANGULAR  HYPERBOLIC  TANGENTS  OF  A  RECTANGULAR  VARIABLE 

Entering  Process 

Let  tanh  (x  +  iy}  be  the  required  function.  Quadrant  the  imaginary  component, 
as  described  under  Tables  VII  and  VIII;  that  is,  divide  y  by  IT/ 2;  so  that  y/(*/2)  —  q. 
The  required  function  is  now  expressed  in  the  form  tanh  (x  +  iq).  Throw  out  mul- 
tiples of  2  from  q  and  retain  only  the  remainder  as  q.  Enter  Table  IX  with  (x  +  iq), 
and  find  the  result  directly  as  u  ±  iv.  It  is  a  well-known  property  of  tanh  (x  +  iy), 
that  it  is  periodic  in  iy,  and  that  the  period  is  i.ir  circular  radians;  or,  in  quadrants, 
i.2.  That  is 

tanh  I  x  +  *  (y  -f  mr)  \  =  tanh  (x  +  iy).  (79) 

where  n  is  any  integer;  or,  in  quadrant  measure  of  the  imaginary, 

tanh  \  x  +  i  (q  +  2n)  j  =  tanh  (x  +  iq).  (80) 

Example:  Required  tanh  (0.25  +  *3o)         =  tanh  (x  +  iy). 

Quadranting,  tanh  (0.25  +  i  19.099)  =  tanh  (x  +  iq). 

Rejecting  multiples  of  2,  tanh  (0.25  +  i  1.099)     =  tan^  (x  +  *'?)• 
We  now  enter  Table  IX  with  x  =  0.25  and  q  =  1.099,  the  nearest  entry  to  which 
x  =  0.25,  q  =  i.i.      The  result  is  2.95122  —  i  1.75011. 

[201] 


EXPLANATORY  TEXT 


INTERPOLATION 


Interpolation  may  be  approximately  effected  by  simple  proportion,  first  in  x  and  then 
in  q,  as  indicated  in  connection  with  Tables  VII  and  VIII;  or,  when  a  higher  degree  of 
precision  is  required,  recourse  may  be  had  to  Taylor's  theorem  in  the  following  form:  — 

tanh  (0  +  A0)  =  tanh  6  +  A0  sech2  0  -  ?—f-  2  sech2  0  tanh  0 


+         -  2  sech2  0  (2  tanh2  0  -  sech2  0)  +   .  .  .  (81) 

o  * 
or 


tanh  i  (*  +  iy)  +  (A*  +  iAy)  \  =  tanh  (x  +  iy)  + 
(A*  +  iAy)'  .  _  _  • 


cosh2  (as  +  « 
and  quadranting, 

tanh  \  (x  +  iq)  +  (A*  +  i&q)  \  =  tanh  (x  +  iq)  + 


-  . 

cosh2  (x  +  i 

so  that  as  far  as  the  second  correction  term:  — 

tanh  \  (x  +  iq)  +  (A*  +  *Ag)  }  =  tanh  (*  +  iq)  +  cosh2 


tanh  (,  +  {„)+  ...  (83) 

\IQ 


(A'O)*  tanh  (ag  +  iq) 
cosh2  (*  +  iq) 

where  A'0  =  (A*  +  iAy)  =  \Ax  +  iAq  (ir/z)  }  (85) 


Example:  Required  tanh  (0.5  +  3'  0.5  5)  =  tanh  (a:  +  ^) 

having  given  cosh  (0.5  +  ^'0.5)  =  0.79735  +  *  0.3684  7  by  Table  VIII 

=  0.87837  /24°.8o3  by  Table  XI 
and  tanh  (0.5  +  ^'0.5)  =  0.76159  -f-  i  0.64805  by  Table  IX 

=  i.  0/40°.  395  by  Table  XII. 
Here  A'0  =  (o  +  ^'0.05  X  1.5708)  =  (o  +  ^'0.07854). 

*'  0.07854  ,    0-00617  X  i  /4Q°. 

tanh  (0.5  +  .0^5)  =  0.76x59  +  .0.64805  +  2^        + 


=  0.76159  +  i  0.64805  + 

0.77153  0.77153 

=  0.76159  +  i  0.64805  +  0.10180/40°. 394  +  o.oo8oo\9°.2ii 
=  0.76159  +  i  0.64805  +  0.07753  +  *  0.06597  ~J~  0.00790  —  i  0.00128 
=  0.84702  +  i  0.71274. 
The  correct  value  is     0.84752  +  *  0.71229. 

As  the  third  correction  term  is  inconvenient  for  computation,  it  is  often  preferable 
to  obtain  a  precise  interpolation  of  tanh  (x  +  iy)  in  working  out  the  correct  inter- 
polations of  sinh  (x  +  iy)  and  cosh  (x  +  iy)  by  the  methods  already  illustrated,  and 
then  to  take  their  ratio. 

[202] 


EXPLANATORY  TEXT 

EFFECTS  OF  CHANGES  OF  SIGN  IN  THE  ENTERING  QUANTITY 

If  tanh  (      x  +  iy)  u  +  iv  (86) 

then  tanh  (      x  —  iy)  =       u  —  iv  (87) 

tanh  (—  x  +  iy)  =  —  u  +  iv  (88) 

and  tanh  (—  x  —  iy)  =  —  (u  +  iv).  (89) 

Consequently,  changes  in  the  sign  of  the  entering  quantity  produce  corresponding  changes 
of  sign  in  the  result. 


FIG.  32.  —  Graphs  of  x  +  iy  and  tanh  (*  +  iy)  in  the  XY  and  UV  planes  respectively. 

CIRCULAR  TANGENTS  OF  COMPLEX  "  ANGLES  " 

Since  tan  6  =  —  ita.nh  (id),  (go) 

It  follows  that  tan  (x  +  iy)  =  —  stanh  (—  y  +  ix)  (91) 

or,  quadranting  the  imaginary:  — 

tan  (x  +  iy)  =  —  i  tanh  (—  y  +  i  2x/ir)  =  —  i  tanh  (—  y  +  iq) 
=  —  i  (—  u  +  iv) 
=  v  +  iu.  (92) 

Consequently,  to  find  tan  (x  —  iy)  from  Table  IX,  enter  it  with  y  as  x  and  with  x/(ir/2) 
as  q.     Invert  the  components  of  the  result  and  the  required  function  is  obtained.     Thus 

[•203] 


EXPLANATORY  TEXT 

required  tan  (i+  2*  2).      We  enter  with  x  =  2  and  q  =  0.6366.     The  nearest  entry  is 
x  —  2.0  and  q  =  0.65  for  which  u  +  iv  =  1.01623  +  i  0.03318.      Therefore,  inverting, 

tan  (i  +  i  2)  =  0.03318  +  i  1.01623 
neglecting  the  interpolation  from  q  =  0.65  to  q  =  0.6366. 

GRAPHIC  INTERPOLATION  BY  MEANS  or  CHARTS  IXA,  IXs,  AND  IXc 

These  charts  contain  all  of  the  entries  in  Table  IX,  and  also  a  certain  number  of 
additional  results.  They  present  circles  intersecting  circles  orthotomically;  i.e.,  by 
rectangular  intersection.  It  is  clear  that  for  values  of  x  less  than  o.io,  the  curves  run 
off  Chart  IXA.  In  fact  the  first  curve  shown  of  x  =  o.oi  extends  as  far  as  u  =  100. 
By  taking  x  small  enough,  the  corresponding  values  of  u  and  v  may  become  indefinitely 
great.  The  entire  UV  plane  is  covered  to  infinity  once  between  x  =  o  and  x  =  o°, 
q  =  o  and  q  =  ^  It  is  covered  once  more  for  each  2  quadrants  increase  in  q. 

When  entering  for  tanh  (—  x  ±  iq);  or  for  the  inverse  operation  tanh"1  (—  u  ±  »), 
it  must  be  remembered  that  the  confocal  conic-section  diagrams  VII  and  VIII 
are  complete  for  negative  as  well  as  for  positive  values  of  x  and  q;  but  that  only  half 
of  the  UV  plane  is  presented  in  Charts  IX.  The  full  graph  is  indicated  in  Fig.  32, 
by  the  aid  of  which  the  functions  corresponding  to  negative  real  values  are  readily 
apprehended. 

TABLE  X 

sinh  (x  +  iq)  =  r  /y 
POLAR  HYPERBOLIC  SINES  or  A  RECTANGULAR  VARIABLE 

This  table  corresponds  completely  to  Table  VII,  already  considered,  except  that  it 
offers  results  in  polar  instead  of  rectangular  coordinates. 

To  find  sinh  (x  +  iy)  expressed  in  polar  coordinates,  quadrant  the  imaginary,  and 
express  the  entering  variable  as  (x  +  iq).  Reject  multiples  of  4  in  q,  and  if  the  re- 
mainder exceeds  2,  reject  2  but  change  the  sign  of  the  total  result. 

INTERPOLATION  BY  SIMPLE  PROPORTION 
Required  sinh  (o.  1  5  +  i  0.25)  having  given 

sinh  (0.2  +  1*0.2)  =      o.36882/58°.723.         sinh  (0.2  +  20.3)  =  0.49663/68°.  825. 
sinh  (o.i  +  20.2)  =      0.32485  /72°.947.         sinh  (o.i  +  ^'0.3)  =  0.46491  778°.  932. 

Diff.  for  o.i  x    =  +0.043977  —  14°.  224.  Diff.  for  o.i  x  =  +0.031727  —  10°.  107. 

Diff.  for  0.05  x  =      0.02I99/—   7°.  112.          Diff.  for  0.05  x  =      0.015867—   5°.Q54. 


sinh  (0.15  +  foj)  =  o.48o77/73°.878. 
sinh  (0.15  +  1*0.2)  =      0.34684  /65°.83S.        sinh  (0.15  +  1*0.2)  =  0.34684/65°.  835. 

Diff.  for  i  o.i    =  0.133937  8°-°43- 
Diff.  for  2*0.015  =  o.  06697  /  4°.  022. 


sinh  (0.15  +  2*0.25)  =  0.41381/69°. 857. 
Correct  Value  0.41 124/70°. 229. 

[  204] 


EXPLANATORY  TEXT 

INTERPOLATION  BY  TAYLOR'S  THEOREM 

For  a  higher  degree  of  precision  than  is  obtainable  by  simple  proportion,  it  is  con- 
venient to  use  rectangular  coordinates  and  apply  formula  (62).  Thus,  required  sinh 
(0.15  +  20.25).  Here  referring  to  Table  VII  and  to  the  work  on  page  195,  we  find 
for  the  result  u  +  iv  =  0.13911  +  i  0.38701. 

Here        ^0.38701  =  1.5877110         yo0.^'.^  =  jo°.22g         ^0.13911  =  1.1433584 
log  0.13911  =  1.1433584  log  sec.  7o°.229  =  0.4707400 

0.4443526  log  0.41124=  1.6140984 

log  tan  70°. 13'  =  0.4440674       Result  0.41124/70°. 229. 

2853       Correct  Value  0.41 124  /70°.229. 
0-72  3968 

INTERPOLATION  BY  CHARTS  X-XlA  AND  X-XIs 

These  charts  present  the  polar  coordinate  results  on  rectangular  coordinate  sheets, 
so  that  they  are  not  true  graphs,  but  are  merely  to  be  regarded  as  interpolation  dia- 
grams. 

To  find  sinh  (x  +  fy),  proceed  as  in  the  use  of  the  tables  and  quadrant  the  imaginary 
so  as  to  obtain  the  entering  quantity  in  the  form  (x  +  iq}>  Enter  with  the  curvilinear 
coordinates,  taking  the  more  nearly  vertical  wavy  lines  for  x  and  the  more  nearly  hori- 
zontal lines  for  q,  starting  from  the  line  SS  as  the  zero  of  q.  Read  off  the  result  on 
the  rectangular  background  to  the  left-hand  scale  of  ordinates. 

When  we  leave  X-XlA  and  enter  X-XlB,it  is  noticeable  that  the  curves  of  constant 
x  approach  vertical  straight  lines  and  the  curves  of  constant  q  approach  horizontal 
straight  lines.  At  and  beyond  x  =  3.0,  we  may  approximate  to  the  modulus  r  at  any 
required  q,  by  taking  the  value  of  r  at  q  =  0.5  and  simple  proportional  parts  between 
this  and  r  at  q  =  o  or  r  at  q  =  i.o.  The  change  in  modulus  r  between  q  =  0.5,  and 
either  of  the  above  limits  is  very  nearly  e~z/2-  Thus  at  x  =  3.5  and  q  =  0.5,  r  by  the 
tables  is  16.55774.  At  q  =  o,  r  =  16.54263,  a  change  of  —  0.01511,  and  at  q  =  i.o 
r  =  16.57282,  a  change  of  +  0.01508.  The  value  of  c~3-6/2  will  be  found  to  be  0.01510, 
and  over  the  entire  range  of  q  from  o  to  i.o,  the  change  in  r  follows  in  nearly  simple 
proportion. 

Beyond  x  =  3.2,  the  limit  of  Chart  X-XIs,  the  values  of  r  can  be  obtained 
by  the  above  rule  applied  to  Table  X  and  with  the  aid  of  Chart  VII-VIIIc.  The 
values  of  the  amplitude  y  beyond  x  =  3  closely  approximate  to  q  quadrants.  That 

tx 
is  sinh  (x  -f-  iq)  approximates  to  —  ^£,  with  q  in  quadrant  measure. 

TABLE  XI 

cosh  (x  -\-  iq)  —  *  /7 
POLAR  HYPERBOLIC  COSINES  OF  A  RECTANGULAR  VARIABLE 

Table  XI  corresponds  completely  with  Table  VIII,  except  that  it  gives  results 
expressed  in  polar  instead  of  rectangular  coordinates.  It  is  entered  with  (x  -f  iq}  just 

as  in  Table  VIII. 

[205] 


Interpolation  may  be  effected  by  simple  proportion,  as  in  the  case  of  Table  X,  or 
when  a  higher  degree  of  precision  is  required,  it  may  be  carried  on  by  Taylor's  theorem. 
In  the  latter  case,  it  is  more  convenient  to  refer  to  the  corresponding  entries  in  Table 
VIII,  interpolating  according  to  formula  (716).  The  rectangular  coordinates  duly 
interpolated  are  then  transformed  into  polar  coordinates,  as  in  the  last  example  on 
page  225. 

INTERPOLATION  BY  CHARTS  X-XlA  AND  X-XIs 

When  Charts  X-XI  are  used  to  find  cosh  (x  +  iy) ,  the  imaginary  is  first  quadranted 
by  dividing  with  ?r/2,  so  as  to  obtain  the  entering  variable  in  the  form  cosh  (x  +  iq). 
Starting  then  from  q  =  o  at  the  horizontal  line  CC,  near  the  middle  of  the  chart,  the 
underscored  figures  correspond  to  q  for  a  little  more  than  the  first  quadrant.  The 
manifest  repetition  of  the  curves  enables  the  lower  half  of  the  sheet,  however,  to  be 
used  for  the  second  quadrant.  The  result  is  read  off  on  the  rectangular  background 
to  the  right-hand  scale  of  argument. 

Beyond  x  =  3.2,  the  limit  of  Chart  X-XlB  reference  may  be  had  to  Chart  VII- 
VIIlB;  or  the  approximate  formula  may  be  used: 

ex 
cosh  (x  +  iq}  =  —  /£.  (93) 

the  argument  q  of  the  result  being  interpreted  in  quadrant  measure  and  converted  into 
degrees. 

TABLE   XII 

tanh  (x  +  iq)  =  r  /y 
POLAR  HYPERBOLIC  TANGENTS  OF  A  RECTANGULAR  VARIABLE 

Table  XII  corresponds  completely  with  Table  IX,  except  that  it  gives  results 
expressed  in  polar  instead  of  rectangular  coordinates. 

If  we  desire  to  find  tanh  (x  +  iy),  we  must  first  divide  y  by  IT/ 2  so  as  to  obtain  the 
entering  quantity  in  the  form  (x  +  iq).  Multiples  of  2  are  then  rejected  in  q  leaving 
a  remainder  less  than  2.  With  this  remainder  the  table  is  entered. 

Interpolation  may  be  made  by  simple  proportion  to  a  moderate  degree  of  precision. 

GRAPHIC  INTERPOLATION  BY  MEANS  OF  CHARTS  XIlA,  B,  c,  D 

These  charts  cover  between  them  the  full  range  of  Table  XII.  To  find  tanh  (x  +  iq) 
from  them  with  q  less  than  2,  find  the  proper  chart,  and  enter  orr'the  curvilinear  co- 
ordinates keeping  the  underscored  number  for  q.  Read  off  the  result  on  the  rectilinear 
background. 

For  tan  (x  +  iy)  and  also  for  the  effects  of  changes  of  sign,  see  directions  in  the  dis- 
cussion on  Table  IX. 

To  find  tanh""1  (r/r),  enter  immediately  on  the  rectangular  background  of  r  and  7 
in  the  proper  chart,  and  read  off  at  the  correct  intersection  the  corresponding  values 
on  the  curvilinear  coordinates.  The  result  will  appear  in  terms  of  (x  +  iq).  The 
imaginary  q  must  be  dequadranted,  or  multiplied  by  7r/2,  in  order  to  be  expressed  in 

terms  of  (x  +  iy). 

[  206] 


EXPLANATORY  TEXT 

TABLE   XIII 

/  (4  +  iq)  =  u  +  iv  or  r  /y 
RECTANGULAR  AND  POLAR  FUNCTIONS  OF  THE  RECTANGULAR  VARIABLE  (4  +  iq) 

In  this  table  the  hyperbolic  sine,  cosine  and  tangent  of  (4  +  iq)  are  collected  from 
q  —  o  to  q  =  2.0.  The  results  are  expressed  both  in  rectangular  coordinates  (u  +  iv), 
and  in  polar  coordinates  r  /y. 

It  will  be  seen  that  the  moduli  of  the  tangents  vary  between  0.99933  and  1.00067, 
or  differ  from  unity  by  two  thirds  of  one  per  mil,  at  most.  The  arguments  also  differ 
from  o°  by  less  than  0.04°,  or  about  2'.  17"  of  arc. 

Beyond  x  =  4,  it  is  evident  that  the  hyp.  sine  and  cosine  differ  by  so  small  a  per- 
centage, that  no  tabulation  of  these  differences  would  ordinarily  be  required. 

TABLE   XIV 

e*/2  and  log  10  (ex/2) 
SEMI-EXPONENTIALS 

This  table  enables  the  hyp.  sine  or  cosine  of  any  rectangular  variable  (x  +  iq)  to 
be  found  for  values  of  x  greater  than  4  and  less  than  10.  It  is  shown  in  the  preceding 
table  that  when  x  reaches  4,  the  ratio  of  the  sine  to  the  cosine  never  differs  from  unity 
by  more  than  two-thirds  of  i  per  mil.  This  deviation  from  unity  rapidly  diminishes  as 
x  is  further  increased.  Consequently,  the  sine  and  cosine  may  each  be  computed  from 
the  formula. 

sinh  (x  +  iq)  =  cosh  (x  -f-  iq)  =  —  /q.  (94) 

Example:  Required  the  value  of  sinh  (8.51  -f-  i  25.75).  The  first  step  is  to  quad- 
rant the  imaginary  by  dividing  with  x/2,  as  on  page  191.  This  gives  the  required  func- 
tion in  the  form  sinh  (8.51  +  i  16.393).  Rejecting  multiples  of  4.0  in  q,  we  may  then 
write  it  sinh  (8.51  +  3*0.393).  Turning  to  the  top  of  page  143,  we  find  ex/2  =  2482.082 
for  x  =  8.51;  so  that  the  result  is  2482.082  70.393  quadra-nt.  Expressing  the  argument 
in  .degrees  by  multiplying  with  90  and  we  have  0.393  X  90  =  35.37°.  Thus 

sinh  (8.51  +  i  16.393)  =  cosh  (8.51  +  i  16.393)  =  2482.082  /35°.37. 

INTERPOLATION  IN  x 
.  Since  ^  =  ?^=<-I+Al+(M!  +  +...  (95) 


it  follows  that  when  Ax  is  a  small  quantity,  it  suffices  to  multiply  the  tabular  value  of 
er/2  by  (i  -f  Ax)  in  order  to  arrive  at  the  interpolated  result  unless  (A*)2/  2!  the  second 
correction  term,  is  of  sufficient  magnitude  to  need  consideration. 

[207] 


EXPLANATORY  TEXT 

Example:  To  find  sinh  (8.51  +  iq),  having  given  that  sinh  (8.50  +  iq)  =  2457.383  /£. 
Here  Ax  =  o.oi. 

2457-383  X          1  =  1  2457.383 

2457-383  X    Ax     =  o.oi  24.574 

(A*)2 
2457-383  X  — -—  =  0.00005  -123 

2482.080 

Result  2482. 080  /q. 

Tabulated  value  2482.082  [^ 

TABLE   XV 

/(*  +  *  o) 
REAL  HYPERBOLIC  FUNCTIONS 

This  is  a  short  table  of  real,  as  distinguished  from  complex  hyperbolic  functions  for 
convenience  of  reference.  It  was  prepared  and  published  by  the  author  in  1903  in  re- 
lation to  continuous-current  electric  circuit  applications,  taking  the  sines,  cosines,  and 
tangents  from  Ligowski's  tables,  and  adding  the  corresponding  computed  reciprocals 
for  the  cosecants,  secants,  and  cotangents.  Much  more  extensive  tables  of  real  hyper- 
bolic functions  are,  however,  available.  See  Bibliography,  page  211. 

TABLE   XVI 
SUBDIVISIONS  or  A  DEGREE 

This  is  a  short  table  for  convenience  in  changing  the  expression  of  a  circular  angle 
from  decimals  of  a  degree  to  minutes  and  seconds,  or  inversely.  By  its  aid,  three- 
decimal  subdivisions  of  a  degree  may  be  converted  into  minutes  and  seconds  of  arc,  by 
direct  inspection;  or  minutes  and  seconds  may  be  read  off  as  decimals  of  a  degree  ta 
three-digit  accuracy. 

METHODS  EMPLOYED  IN  COMPUTATION 

Tables  I  to  V,  inclusive,  were  computed  as  one  group,  and  Tables  VII  to  XIII, 
inclusive,  as  a  separate  group. 

Tables  I  to  V  were  computed,  at  first,  by  using  the  formulas:  — 

sinh  (x  +  iy)  =  V  sinh2  x  +  sin2  y  /tan"1  (tan  y  /  tanh  x)  =  ri  /yi.  (96) 

cosh  (x  +  iy)  =  Vcosh2#  —  sin2;y  /tan"1  (tan  y  •  tanh  x)  =  r2  /72.  (97) 

tanh  (x  +  iy)  =  (ri/r2)  /TI  ~  72-  (98) 

At  a  later  stage  of  the  work,  the  following  formulas,  kindly  suggested  by  Professor 
Bouton,  were  substituted:  — 

sinh  (x  +  iy)  =  Vcosh  2X  •  sin  z  /tan"1  (tan  y  /  tanh  x)  =  ri  fji.  (99) 

cosh  (x  +  iy)  =  V  cosh  sx  •  cos  z  /tan"1  (tan  y  •  tanh  x)  =  r%  /jz.  (100) 

tanh  (x  +  iy)  =  tan  z  /y\  —  72-  (101) 

[208] 


EXPLANATORY  TEXT 

Where  the  auxiliary  circular  angle  z  is  defined  by : 

cos  2y 

-    =   COS   2Z.  (102) 

cosh  2X 

The  arithmetical  work  was  conducted  with  the  aid  of  five-place  logarithms,  and  was 
checked  by  tabulating  successive  first  and  second  differences  in  the  tabulated  results. 
Tables  VII  to  XII  were  computed  by  means  of  the  following  formulas:  — 

sinh  (x  +  iy)  =  sinh  x  cos  y  -f-  i  cosh  x  sin  y.  (103) 

cosh  (x  -f-  iy)  =  cosh  x  cos  y  +  *  sinh  x  sin  y.  (104) 

,    f         .  .  sinh  2X  sin  2y 

tanh  (x  +  ty)  =  — r- —        -  -f  *  — —     — ^ (105) 

cosh  2#  -f  cos  2y         cosh  2*  +  cos  2y 

A  standard  schedule  was  prepared  and  seven-place  logarithms  used  in  the  compu- 
tation. The  value  of  tanh  (x  +  iy)  was  arrived  at  in  two  ways,  first  by  dividing  (103) 
by  (104),  and  second  by  the  independent  formula  (105).  If  these  two  methods  did  not 
give  identical  results  for  tanh  (x  -f-  iy)  to  five  decimal  places,  when  expressed  both  in 
rectangular  and  polar  coordinates,  the  steps  of  the  computation  were  gone  over  afresh.* 
Complete  agreement  being  secured,  leads  to  the  inference  that  the  values  of  sinh,  cosh, 
and  tanh  (x  +  iy)  are  correct,  at  least  as  far  as  their  logarithms. 

Finally,  all  of  the  tables  have  been  reduced  to  graphic  form  in  the  Atlas,  each  entry 
of  the  tables  being  marked  off  on  its  proper  chart  with  a  sharp  needle,  and  the  ruling 
pen  drawn  through  the  successive  punctures.  In  this  process  a  certain  number  of 
errors  were  discovered  and  rectified.  The  tables  were  then  set  up  in  type  from  the 
MSS.  used  in  making  the  charts,  and  were  proofread  three  times.  By  this  procedure 
it  is  hoped  that  the  outstanding  errors  are  neither  large  nor  numerous. 

BIBLIOGRAPHY  AND  APPLICATIONS  OF  HYPERBOLIC  FUNCTIONS 

Hyperbolic  functions  of  a  real  variable  are  employed  extensively  in  mathematics 
generally.  In  particular,  they  are  used  in  the  solution  of  cubic  equations. 

In  navigation,  real  hyperbolic  functions  enter  in  connection  with  Mercator  sailing. 

In  cartography,  real  hyperbolic  functions  are  used  in  preparing  maps  on  certain  pro- 
jections, especially  on  Mercator's  projection,  which  appears  to  have  been  the  first 
application  of  hyperbolic  functions. 

In  statics,  real  hyperbolic  functions  naturally  present  themselves  in  relation  to  the 
properties  of  the  catenary  and  of  the  funicular  polygon;  also  in  the  discussion  of  the 
forms  and  stresses  of  elastic  bodies. 

In  dynamics,  the  same  functions  present  themselves  in  the  theory  of  vibrations, 
and  in  the  motion  of  bodies  through  a  resisting  medium. 

*  The  author  desires  to  express  his  acknowledgement  of  the  care  and  painstaking  effort  of  his  assistants 
engaged  in  computation,  namely, 

Miss  Ethel  Smith,  A.B.  Radcliffe,  1911. 
Miss  A.  F.  Daniell,  A.B.  Radcliffe,  1911. 
Miss  Mary  M.  Devlin,  A.B.  Radcliffe,  1912. 
Miss  Hope  M.  Hearn,  A.B.  Radcliffe,  1912. 
C  209] 


EXPLANATORY   TEXT 

A  good  summary  of  the  historical  development  of  real  hyperbolic  functions  is  given 
in  Becker  and  van  Orstrand's  "Hyperbolic  Functions,"  Smithsonian  Mathematical 
Tables,  1909,  together  with  a  fine  compendium  of  formulas  involving  these  functions.* 

In  electrical  engineering,  the  earliest  published  application  of  real  hyperbolic  func- 
tions is  perhaps  in  T.  H.  Blakesley's  "Alternating  Currents  of  Electricity,"  London, 
1889,  which  also  appends  a  short  table  of  these  (real)  functions.  The  real  functions 
were  also  introduced  by  Sir  J.  J.  Thomson,  in  "The  Electrician,"  Vol.  XXVIII,  page 
599,  1891.  "On  the  Heat  Produced  by  Eddy  Currents  in  an  Iron  Plate  Exposed  to  an 
Alternating  Magnetic  Field." 

The  fundamental  differential  equation  of  the  alternating  potential-current,  steady- 
state  distribution  along  a  uniform  conductor,  involving  hyperbolic  functions,  nominally 
real,  seems  to  have  been  first  published  by  O.  Heaviside  in  1893,  "Electromagnetic 
Theory,"  Vol.  I,  page  450. 

The  first  published  application  of  complex  hyperbolic  functions  to  the  last-named 
problem  was  by  the  author,  "On  the  Fall  of  Pressure  in  Long-Distance  Alternating- 
Current  Conductors,"  Electrical  World,  N.  Y.,  Vol.  XXIII,  page  17,  January,  1894 
and  "The  Electrician,"  London  (abstract),  Vol.  XXXII,  page  239,  January  5,  1894. 

Complex  hyperbolic  functions  also  present  themselves  in  the  discussion  of  Hertzian- 
wave  reflections,  and  in  other  branches  of  electrical  engineering.  They  naturally  enter 
the  subject  of  confocal  ellipses  and  hyperbolas,  such  as  Captain  Weir's  Azimuth  diagram 
of  these  confocals,  for  indicating  the  azimuth  of  a  celestial  object  in  terms  of  the  hour- 
angle,  latitude  and  declination.  (Godfray's  "Astronomy,"  §  222.) 

The  mathematical  discussion  of  hyperbolic  functions  is  found  in  GreenhilTs  "Differ- 
ential and  Integral  Calculus,"  Macmillan  and  Co.,  1896;  Ligowski's  "Tafeln  der  Hyper- 
belfunctionen  und  der  Kreisfunctionen,"  Berlin,  Ernst  &  Korn,  1890;  McMahon's 
"Hyperbolic  Functions,"  Wiley  and  Sons,  N.  Y.,  1896;  Becker  and  van  Orstrand's 
"Hyperbolic  Functions,"  Smithsonian  Institution,  1909;  Vassall's  "Nouvelles  Tables 
des  Logarithmes,"  Paris,  Gauthier-Villars,  1872;  as  well  as  other  text-books. 

Works  dealing  with  the  applications  of  hyperbolic  functions  to  electrical  engineering 
are:  "The  Application  of  Hyperbolic  Functions  to  Electrical  Engineering  Problems," 
by  the  author,  The  University  of  London  Press,  1911,  and  Fleming's  "The  Propagation 
of  Electrical  Currents  in  Telephone  and  Telegraph  Conductors,"  Constable  &  Co., 
London,  1911. 

A  three-dimensional  complex-angle  geometrical  model,f  from  which  the  hyperbolic 
sines  and  cosines  of  complex  angles  can  be  presented  projectively,  has 'been  constructed 
and  described. 

BRIEF  BIBLIOGRAPHY  or  TABLES  or  HYPERBOLIC  FUNCTIONS 

(i)  "Tafeln  der  Hyperbelfunctionen  und  der  Kreisfunctionen,"  by  Dr.  W.  Ligowski, 
Berlin,  1890,  Ernst  and  Korn,  104  pages,  giving  five-figure  logarithms  of  sinh  6,  cosh  6, 
and  tanh  0  up  to  8  =  9,  by  steps  of  o.ooi  up  to  6  =  2  and  from  2.0  to  9.0  by  steps  of 

*  This  compendium  has,  by  permission,  been  included  in  this  book  at  its  second  edition,  as  Table  XXIH. 
f  "A  new  geometrical  model  for  the  orthogonal  projection  of  the  cosines  and  sines  of  complex  angles"  by 
A.  E.  Kennelly,  Proc.  Am.  Ac.  of  Arts  and  Sciences,  Vol.  54,  April,  1919,  pp.  371-378. 

[210] 


EXPLANATORY  TEXT 

o.oi;  also  the  Gudermannian  angle  to  two  or  more  decimals  of  a  second  of  arc,  and 
other  tables. 

(2)  Smithsonian    Mathematical    Tables,    "Hyperbolic    Functions,"    by    George    F. 
Becker,  and  C.  E.  van  Or  strand,   Smithsonian  Institute,  Washington,  D.C.,  1909,  321 
pages,  giving  five-figure  logarithms  of  sinh  6,  cosh  0,  and  tanh  0,  by  steps  of  o.oooi 
up  to  o.i,  by  steps  of  o.ooi  from  o.i  to  3.0,  and  by  steps  of  o.oi  from  3.0  to  6.0;  also 
similar  five-figure  tables  of  natural  real  hyperbolic  functions,  and  various  other  tables. 

(3)  "Alternating-Current  Phenomena  in  Parallel  Conductors,"  Vol.  I  by  F.  E.  Pernot, 
John  Wiley,  New  York,  1918,  containing  a  Table  of  six-decimal  logarithms  of  hyperbolic 
functions,  up  to  2.0  by  steps  of  o.ooi.    These  present  a  higher  order  of  precision  by 
one  unit,  than  have  been  previously  available  for  real  hyperbolic  functions. 


The  following  is  a  list  of  all  the  tables  of  Complex  Hyperbolic  Functions  known 
to  the  present  writer,  in  the  order  of  date  of  publication:  — 

(4)  ChrystaFs   "Algebra,"   Edinburgh,    1889,   briefly  discusses  the   theory  of  sinh  6, 
cosh0,  and  tanh  6  where  6  is  complex;  or  of  the  form  x  +  iy.    Graphs  are  given  in  out- 
line for  these  functions,  from  which  a  few  numerical  values  may  be  read. 

(5)  The   paper   on   "Resonance  in   Alternating-Current  Lines,"   by   E.   J.   Houston 
and  A.  E.  Kennelly,  Transactions  A.  I.  E.  E.,  April,  1895,  Vol.  XII,  pages  133-169, 
contains  a  Plate  for  the  graphical  evaluation  of  sinh  6  and  of  cosh  0,  0  being  a  complex 
variable  x  +  iq,  between  the  limits  of  x  =  o  and  #=i.25;g  =  o  and  q  =  «,  by  steps 
of  0.05  in  x  and  q.     The  Plate  is  40  cm.  X  34  cm.  and  corresponds  to  Plates  VII — 
VIILv  of  the  Atlas  prepared  from  tables  in  this  book,  except  that  it  gives  the  result  in 
regular  polar  coordinates  instead  of  regular  rectangular  coordinates.     It  was  produced, 
by  a  graphical  process,  for  a  precision  of  the  2.5th  order. 

(6)  The  first  tables  of  complex  hyperbolic  functions  were  a  short  set  published  by 
Dr.  James  McMahon  in  his  Chapter  IV,  entitled  "Hyperbolic  Functions,"  of  a  book 
by  Merriam  and  Woodward  on  "Higher  Mathematics,"  pages  107-168.     The  tables 
gave  sinh  (x  +  iy)  and  cosh  (x  +  iy)  from  x  =  o  to  x  =  1.5,  by  steps  of  o.i,  and  also 
from  y  =  o  to  y  =  1.5,  by  steps  of  o.i,  Wiley  &  Sons,  New  York,  1896.    The  chapter 
has  since  been  issued  as  a  separate  volume  by  the  same  publishers. 

(7)  A  table  of  hyperbolic  functions  of  semi-Lmaginaries  or  sinh,  cosh,  tanh,  coth, 
sech  and  cosech  of  x /45°,  by  steps  of  o.i  in  x  up  to  x  =  20.5,  was  published  by  the 
present  writer  in  a  paper  on  "The  Alternating-Current  Theory  of  Transmission  Speed 
over  Submarine  Telegraph  Cables,"  in  the  Proceedings  of  The  International  Electrical 
Congress  of  St.  Louis,  Section  A,  Vol.  I,  pages  68-105,  1904.    This  table  is  reproduced 
in  Table  VI  of  this  volume. 

(8)  Some  short  tables  of  sinh,  cosh,  tanh,  coth,  sech,  and  cosech  p/5  by  steps  of  o.i 
in  p,  up  to  p  =  1.5,  for  five  particular  values  of  5,  published  by  the  present  writer  in  an 
article  on  "The  Distribution  of  Pressure  and  Current  over  Alternating-Current  Cir- 
cuits," in  the  Harvard  Engineering  Journal,  1905-06. 

C"!  3 


EXPLANATORY   TEXT 

(9)  Short  three-digit  tables  of  sinh  and  cosh  (x  +  iy)  up  to  x  =  i,  and  y  =  i,  by 
W.  E.  Miller,  in  a  paper  "Formulae,  Constants,  and  Hyperbolic  Functions  for  Trans- 
mission-Line Problems"   in   the  General  Electrical  Review,   Schenectady,   N.   Y.,   May, 
1910.    Supplement. 

(10)  "Tables   of   Hyperbolic   Functions   in    Reference    to   Long   Alternating-Current 
Transmission  Lines,"  published  by  the  present  writer  in  the  Transactions  of  the  American 
Institute  of  Electrical  Engineers,   December   1911,  pages   2495-2506.     These  give  sinh, 
cosh,  and  tanh  p  /5  from  p  =  o  to  p  =  0.5,  by  steps  of  o.i,  and  from  5  =  60°  to  5  =  90° 
by  steps  of  i°.    These  tables  are  incorporated  in  Tables  I,  II,  and  III  of  this  volume. 

(n)  "Tables  of  Sines,  Cosines,  Tangents,  Cosecants,  Secants,  and  Cotangents  of 
Real  and  Complex  Hyperbolic  Angles,"  published  by  the  present  writer  in  The  Har- 
vard Engineering  Journal"  1912.  These  gave  sinh,  cosh,  and  tanh  p /8  from  p  =  o 
to  p  —  i  by  steps  of  o.i,  and  from  5  =  45°  to  5  =  90°  by  steps  of  i°;  also  corresponding 
tables  of  (sinh  0)/0  and  of  (tanh  0)/0.  These  tables  are  published  in  separate  form  by 
the  Harvard  Engineering  Journal.  They  are  incorporated  in  tables  I,  II,  III,  IV,  and  V 
of  this  volume. 

NEW  TABLES  INTRODUCED  IN  THE  SECOND  EDITION 

Tables  I  to  V  in  this  volume  were  computed  for  the  range  of  45°  to  90°  in  the  slope 
or  argument  5  of  the  entering  vector  quantity;  because  at  that  time  it  did  not  appear 
that  there  would  be  any  need  for  the  range  from  o°  to  45°.  Alternating-current  lines 
used  for  the  transmission  or  distribution  of  power  have  linear  hyperbolic  angles  a,  the 
slope  of  which  is  commonly  between  80°  and  90°,  rarely  falling  as  low  as  45°.  It  has 
been  shown  during  recent  years,  however,  that  railway-signal  engineers  employ  track- 
signaling  circuits,  formed  of  the  rails.  These  are  metallic  circuits  of  low  frequency, 
small  linear  capacitance  and  large  distributed  linear  leakance.  The  linear  hyperbolic 
angles  a  of  such  circuits  develop  slopes  lying  within  the  range  5  =  o°  to  45°.  It  has 
therefore  become  desirable  to  cover  this  range,  at  least  as  far  as  p  =  i.  For  that  pur- 
pose, Tables  XVII  to  XXI  have  been  inserted.  They  run  by  steps  of  0.05  in  p,  from 
o  to  i.o,  and  by  steps  of  5°  in  5,  from  o°  to  45°.  This  new  tabulated  material  is 
available  for  use  in  track-signaling  and  similar  computations.  It  is  hoped  to  incorpo- 
rate it  graphically  into  the  associated  Chart  Atlas  at  the  first  opportunity. 

Table  XVII  presents  sinh  p/£  as  a  polar  planevector.  It  corresponds  to  and  may 
be  regarded  as  an  extension  of  Table  I.  Similarly  Tables  XVIII,  XLX,  XX  and  XXI 
correspond  respectively  to  Tables  II,  III,  IV  and  V.  Whereas,  however,  Tables  I  to  V 
are  carried  to  five  decimal  places  in  the  sizes  and  three  decimal  places  in  the  slopes  of 
the  evaluated  quantities,  the  new  tables  are  carried  to  six  decimal  places  in  sizes  and 
four  decimal  places  in  slopes.  They  thus  aim  at  one  higher  order  of  precision.* 

Table  XXII  is  similar  to  Table  XI  and  expands  the  region  covered  by  the  first  six 
entries  of  the  latter  into  a  correspondingly  magnified  field.  It  has  been  found  that  in 

*  The  author  desires  to  express  his  acknowledgment  of  the  painstaking  assistance,  on  these  tables,  by  Miss 
Lillian  L.  Hodgdon,  of  the  Harvard  Observatory  computing  staff. 

[212] 


EXPLANATORY   TEXT 

dealing  with  short  lengths  of  alternating-current  line,  having  negligible  linear  induc- 
tance as  well  as  negligible  linear  leakance,  and  therefore  having  a  semi-imaginary  linear 
angle,  there  is  frequent  need  for  a  magnified  table  of  this  kind.  It  may  be  noted  that 
whereas  Table  VI  expresses  slopes  in  degrees  and  minutes,  Table  XXII  expresses  them 
as  degrees,  and  four-place  decimals  of  a  degree. 

Table  XXIII  is  a  useful  collection  of  238  formulas,  with  a  few  insertions,  taken 
from  Becker  and  van  Orstrand's  book  of  Tables  of  real  hyperbolic  functions,  referred  to 
in  the  footnote  on  page  210. 


TABLE  XVII.    HYPERBOLIC  SINES,    sinh  (P/5)  =  r 


0.05 


0 

0.000000 

0.0000 

0.050021 

o.oooo 

0.100167 

o.oooo 

5 

o.oooooo 

5.0000 

0.050020 

5.0042 

0.100164 

5.0167 

10 

o.oooooo 

IO.OOOO 

0.050020 

10.0081 

0.100157 

10.0325 

IS 

o.oooooo 

15.0000 

0.050019 

15.0119 

0.100144 

15.0478 

20 

0.000000 

20.0000 

0.050016 

20.0153 

0.100128 

20.0614 

25 

o.oooooo 

25.0000 

0.050014 

25.0183 

0.100107 

25.0730 

30 

o.oooooo 

30.0000 

0.050011 

30.0208 

0.100083 

30.0828 

35 

o.oooooo 

35.0000 

0.050006 

35.0222 

0.100057 

35-0897 

40 

o.oooooo 

40.0000 

0.050003 

40.0236 

0.100029 

40.0942 

45 

o.oooooo 

45.0000 

0.05000 

45.0236 

O.IOOOO 

45-0952 

0-35 


0.40 


0-45 


0 

0-357190 

o.oooo 

0.410752 

o.oooo 

0.465342 

o.oooo 

5 

0.357081 

5.2014 

0.410589 

5-2625 

0.465109 

5-3314 

10 

0.35675? 

10.3969 

0.410105 

10.5172 

0.464420 

10.6531 

15 

0.356228 

15.5808 

0.409316 

15-7569 

0.463294 

15-9558 

20 

0-355512 

20.7472 

0.408245 

20.9742 

0.461767 

21.2303 

25 

0.354628 

25.8908 

0.406925 

26.1625 

0.459886 

26.4686 

30 

0.353606 

31.0089 

0.405397 

31.3161 

0.457709 

31.6630 

35 

0.352475 

36.0961 

0.403709 

36-4305 

0.455303 

36.8086 

40 

0.351270 

41-1503 

0.401910 

41.5017 

0.452741 

41.8997 

45 

0.35003 

46.1694 

0.40006 

46.5278 

0.45010 

46.9338 

0.70 

o  o 

o  0.758584  o.oooo 

5  0.757700  5-7875 

10  0.755078  H.5533 

15  0.750800  17-2755 

20  0.745004  22.9344 

25  0.737873  28.5103 

30  0.729632  33.9861 

35  0.720538  39.3467 

40  0.710872  44.5800 

45  0.700934  49-6767 


0-75 

o 

0.822317  o.oooo 
0.821227  5.9000 

0.817993  11.7753 

0.812719   17.6019 

0.805576  23.3567 

0.796787  29.0175 
0.786636  34-5647 

0-775439  39-98ii 
0.763544  45-2525 
0.75I3I7  50-3678 


0.80 
o 

0.888106  o.oooo 

0.886780  6.0192 

0.882843  1 2.0108 

0.876425  17.9481 

0.867734  23.8044 


0.857048 

0.844710 
0.831105 

0.816660 

0.801819 


29-5561 
35.1800 
40.6567 
45-9697 

51.1064 


Examples,    sinh  (0.35  /35°)  =  0.352475  /36°.O96i. 
sinh  (0.80 /£}  =  0.886780 /6°.oi92. 


TABLE  XVII.  HYPERBOLIC  SINES. 

O.I5                    O.2O 

sinh  (P  /5)  =  r  /?. 
0.25 

CONTINUED 
0.30 

0 

0 

0 

0 

o 

O 

0.150563 

o.oooo 

0.201336 

O.OOOO 

0.252612 

0.0000 

0.304520 

O.OOOO 

5 

0.150554 

5-0372 

0.201316 

5.0661 

0-252573 

5-1033 

0.304452 

•  5-1483 

10 

0.150529 

10.0733 

0.201256 

10.1303 

0-252455 

10.2033 

0.304248 

10.2922 

15 

0.150488 

15.0172 

0.201157 

15.1906 

0.252263 

15.2972 

0-303915 

15.4275 

20 

0.150432 

20.1380 

0.201024 

20.2450 

0.252002 

20.3825 

0.303465 

20.5500 

25 

0.150362 

25.1644 

0.200859 

25.2919 

0.251680 

25-4561 

0.302909 

25.6558 

3° 

0.150282 

30.1858 

0.200669 

30.3303 

0.251308 

30.5158 

0.302265 

30.7419 

35 

0.150193 

35.2017 

0.200458 

35-3586 

0.250896 

35-5600 

0.301553 

35.8058 

40 

0.150098 

40.2114 

.  0.200233 

40.3761 

0.250457 

40.5872 

0-300795 

40.8456 

45 

0.15000 

45.2146 

O.20OOO 

45.3817 

0.25001 

45-5965 

0.30001 

45.8592 

o. 

50 

0-55 

0.60 

o, 

65 

0 

0 

o 

o 

o 

0 

0.521095 

O.OOOO 

0.578152 

O.OOOO 

0.636654 

O.OOOO 

0.696748 

O.OOOO 

5 

0.520776 

5.4078 

0.577726 

5-4919 

0.636100 

5-5833 

0.696042 

5.6819 

10 

0.519829 

10.8039 

0.576463 

10.9697 

0.634456 

11.1500 

0.693947 

"•3477 

15 

0.518282 

16.1767 

0.574400 

16.4197 

0.631774 

16.6842 

0.690530 

16.9694 

20 

0.516184 

21.5153 

0.571604 

21.8286 

0.628138 

22.1700 

0.685898 

22.5386 

25 

0.513601 

26.8094 

0.568162 

27.1844 

0.623662 

27.5933 

0.680199 

28.0355 

3° 

0.510612 

32.0503 

0.564179 

32.4764 

0.618485 

32.9414 

0.673609 

33-4450 

35 

0.507309 

37-2305 

0-559779 

37-6956 

0.612769 

38.2036 

0.666335 

38.7539 

40 

0.503794 

42.3439 

0.555099 

42.8344 

0.606689 

43-3705 

0.658601 

43-9525 

45 

0.50017 

47.3872 

0.550279 

47.8880 

0.600432 

48.4369 

0.650644 

49-0330 

0.85 

0.90 

0-95 

I 

.00 

o 

o 

o 

o 

0 

O 

0.956116 

O.OOOO 

1.026517 

O.OOOO 

1.099484 

O.OOOO 

1.175201 

O.OOOO 

5 

0.954520 

6.1447 

1.024615 

6.2767 

1.097239 

6.4147 

1.172573 

6.5592 

IO 

0.949784 

12.2592 

1.018975 

12.5200 

1.090583 

12.7928 

1.164779 

13-0775 

15 

0.942064 

18.3131 

1.009783 

18.6967 

1.079736 

19.0986 

1.152083 

19.5180 

20 

0.931612 

24.2775 

0-997344 

24.7753 

1.065062 

25.2969 

1-134913 

25.8417 

25 

0.918768 

30.1255 

0.982060 

30.7253 

1.047043 

31-3544 

1.113841 

32.0122 

3° 

0.903942 

35-8314 

0.964429 

36.5183 

1.026266 

37.2400 

1.089558 

37-9955 

35 

0.887604 

41-3730 

0.945011 

42.1294 

1.003398 

42.9253 

1.062847 

43-7603 

40 

0.870267 

46.7314 

0.924414 

47-5372 

0.979156 

48.3864 

1.034550 

49.2789 

45 

0.852463 

51.8917 

0.903276 

52.7242 

0.954292 

53-6033 

1.005545 

54-5292 

Examples,    sinh  (0.90/20°)  =  0.997344/2^ 
sinh  (1.0/0°)  =  1.175201/0°. 


TABLE  XVIII.    HYPERBOLIC  COSINES.    cosh(p/5)  =  r/7- 


0.05 


o-3S 


0.40 


0.70 


0-7S 


O.IO 


0 

I.OOOOOO 

o.oooo 

1.001250 

o.oooo 

1.005004 

o.oooo 

•  5 

I.OOOOOO 

o.oooo 

1.001231 

0.0125 

1.004929 

0.0497 

10 

I.OOOOOO 

o.oooo 

1.001175 

0.0244 

1.004703 

0.0978 

15 

I.OOOOOO 

o.oooo 

1.001083 

0.0358 

1.004335 

0.1428 

20 

I.OOOOOO 

o.oooo 

1.000958 

0.0461 

1.003836 

0.1836 

25 

I.OOOOOO 

o.oooo 

1.000804 

0.0547 

1.003221 

0.2189 

3° 

I.OOOOOO 

o.oooo 

1.000626 

0.0619 

1.002507 

0.2478 

35 

I.OOOOOO 

o.oooo 

1.000428 

0.0672 

1.001718 

0.2689 

40 

I.OOOOOO 

o.oooo 

1.000218 

0.0705 

1.000876 

0.2820 

45 

I.OOOOOO 

o.oooo 

I.OOOOO 

0.0716 

I.OOOOI 

0.2864 

0.45 


o 

1.061878 

O.OOOO 

1.081072 

o.oooo 

1.102970 

o.oooo 

5 

1.060965 

0.5861 

1.079886 

0.7567 

i.  101477 

0-9453 

10 

1.058252 

1.1561 

1.076361 

1-4933 

1.097041 

1.8669 

15 

1.053819 

1.6950 

1.070598 

2.1911 

1.089786 

2.7411 

20 

1.047791 

2.1872 

1.062762 

2.8303 

1.079919 

3-5450 

25 

1.040347 

2.6189 

1-053079 

3-3936 

1.067721 

4-2567 

30 

1.031702 

2.9772 

1.041829 

3-8639 

1-053539 

4-8550 

35 

I.O22II2 

3.2506 

1.029339 

4.2264 

1.037782 

5.3208 

40 

I.OII859 

3-4295 

1.015974 

4.4678 

1.020908 

5-6375 

45 

I.OOI25 

3-5071 

1.00213 

4-5784 

1.00341 

5.7906 

0.80 


o 

1.255169 

o.oooo 

1.294683 

o.oooo 

1-337435 

o.oooo 

5 

1.251669 

2.1083 

1.290690 

2-3744 

1.332918 

2.6483 

10 

1.241264 

4-1750 

1.278820 

4.7044 

1.319493 

5-2500 

15 

1.224242 

6.1581 

1.259402 

6-9456 

1-297535 

7-7578 

20 

1.201069 

8.0155 

1.232970 

9.0520 

1.267648 

10.1236 

25 

1.172378 

9.7039 

1.200241 

10.9775 

1.230648 

12.2978 

30 

1.138949 

11.1803 

1.162104 

12.6747 

1.187536 

14.2294 

35 

1.101690 

12.4000 

1.119583 

14.0942 

I-I39459 

15.8644 

40 

1.061615 

13-3186 

1.073820 

15.1850 

1.087691 

17-1458 

45 

1.019823 

13-8911 

1.026048 

15.8947 

1.033602 

18.0136 

Examples,     cosh  (0.10/25°)  =  1.003221  /o°.2i89. 

cosh  (0.75/40°)  =  i.Q7382o/i5°.i850. 


TABLE  XVIII.  HYPERBOLIC  COSINES,  cosh  (p/3)  =  r  /?.  CONTINUED 

O.I5                      O.2O                    O.2S                    0.30 

0 

o 

o 

o 

o 

o 

1.011271 

o.oooo 

1.020067 

o.oooo 

1.031413 

O.OOOO 

1.045339 

o.oooo 

5 

I.OIIIOI 

O.I  I  II 

1.019765 

0.1964 

1.030943 

0.3047 

1.044665 

0.4350 

10 

1.010595 

0.2189 

1.018868 

0.3872 

1.029547 

0.6006 

1.042663 

0.8578 

IS 

1.009769 

0.3203 

1.017404 

0.5664 

1.027265 

0.8794 

1.039390 

1.2567 

20 

1.008648 

0.4119 

1.015415 

0.7292 

1.024167 

1.1328 

1.034944 

1.6200 

25 

1.007265 

0.4914 

1.012961 

0.8703 

1.020343 

1-3533 

1.029455 

1-9375 

3° 

1.005662 

0.5561 

1.  010116 

0.9858 

1.015908 

1-5344 

1.023084 

2.J994 

35 

1.003887 

0.6042 

1.006966 

1.0719 

1.010992 

1.6703 

1.  01602  1 

2-3975 

40 

1.001995 

0.6339 

1.003604 

1.1258 

1.005746 

1-7567 

1.008476 

2-5253 

45 

1.00005 

0.6445 

1.00013 

1.1458 

1.00033 

1.7901 

1.00068 

2-5773 

0.50 

0.55 

0.60 

0.65 

o 

o 

0 

o 

o 

O 

1.127626 

O.OOOO 

1.155101 

O.OOOO 

1.185465 

o.oooo 

1.218793 

o.oooo 

5 

1.125794 

1.1506 

1.152899 

1.3711 

1.182861 

1.6053 

1.215756 

1.8514 

IO 

1.120349 

2.2736 

1.146351 

2.7108 

1.175118 

3-1753 

1.206728 

3.6644 

15 

1.  111445 

3-34" 

1.135641 

3.9872 

1.162452 

4-6750 

1.191956 

5.4000 

20 

1.099330 

4.3264 

1.121066 

5.1694 

1.145211 

6.0689 

1.171848 

7.0194 

25 

1.084345 

5.2028 

1.103034 

6.2267 

1.123873 

7.3222 

1.146955 

8.4836 

30 

1.066914 

5-9453 

1.082045 

7.1292 

1.099026 

8.4006 

1.117959 

9-7533 

35 

1.047534 

6.5297 

1.058692 

7.8481 

1.071363 

9.2700 

1.085656 

10.7894 

40 

1.026760 

6-9350 

1-033637 

8.3566 

1.041655 

9.8980 

1.050938 

"•5539 

45 

1.00520 

7.1424 

1.007598 

8.6311 

1.010745 

10.2544 

1.014773 

12.0094 

0.85 

0.90 

0-95 

1.  00 

0 

o 

0 

0 

o 

O 

1-383531 

o.oooo 

1.433086 

o.oooo 

1.486225 

o.oooo 

1.543081 

0.0000 

5 

1.378460 

2.9289 

1.427430 

3-2150 

1.479951 

3.5053 

1.536155 

3.7989 

IO 

1-363390 

5.8092 

1.410623 

6-3794 

1-461313 

6.9589 

1.515588 

7-5475 

15 

1.338746 

8.5914 

1-383145 

9.4428 

1.430851 

10.3086 

1.481987 

11.1858 

20 

1.305212 

11.2253 

1-345771 

12.3525 

1.389438 

13.5005 

1.436333 

14-6653 

25 

1.263709 

13-6589 

•299533 

I5-0545 

1-338233 

16.4789 

1.379926 

17.9267 

3° 

1-215358 

15-8375 

.245685 

17.4914 

1.278634 

19.1844 

1.314321 

20.9097 

35 

1.161440 

17.7030 

-185653 

19.6019 

1.212219 

21-5531 

1.241261 

23-5483 

40 

1.103367 

19.1936 

.120988 

21.3192 

1.140689 

23-5142 

1.162611 

25.7689 

45 

1.042645 

20.2411 

•053338 

22.5686 

1.065840 

24.9872 

1.080307 

27.4870 

Examples,    cosh  (0.25/30°)  =  1.015908/1" 
cosh  (1.00/40°)  =  1.162611/2.5 


[2173 


TABLE  XIX.    HYPERBOLIC  TANGENTS,    tanh  (p  l&)  =  r 


0.05 


o 

o.oooooo 

o.oooo 

0.049958 

o.oooo 

0.099668 

o.oooo 

5 

o.oooooo 

5.0000 

0.049959 

4.9917 

0.099673 

4.9670 

10 

o.oooooo 

IO.OOOO 

0.049961 

9-9837 

0.099688 

9-9347 

IS 

o.oooooo 

15.0000 

0.049965 

14.9761 

0.099712 

14.9050 

20 

o.oooooo 

20.0000 

0.049968 

19.9692 

0.099745 

19.8778 

25 

o.oooooo 

25.0000 

0.049974 

24.9636 

0.099786 

24.8541 

3° 

o.oooooo 

300000 

0.049980 

29.9589 

0.099833 

29.8350 

35 

O.OOOOOO 

35.0000 

0.040985 

34-9550 

0.099886 

34.8208 

40 

o.oooooo 

40.0000 

0.049992 

39-9531 

0.099941 

39.8122 

45 

o.oooooo 

45.0000 

0.05000 

44.9520 

O.IOOOO 

44.8087 

o-35 


0.40 


0.70 


o-75 


0-45 


0 

0.336375 

o.oooo 

0-379949 

o.oooo 

0.421899 

o.oooo 

s 

0.336562 

4-6153 

0.380215 

4-5058 

0.422260 

4.3861 

10 

0.337H9 

9.2408 

0.381011 

9.0239 

0.423339 

8.7862 

15 

0.338036 

13-8858 

0.382325 

13.5658 

0.425124 

13-2147 

20 

0.339296 

18.5600 

0-384135 

18.1439 

0.427594 

17.6853 

25 

0.340875 

23.2719 

0.386414 

22.7689 

0.430717 

22.2119 

3° 

0.342740 

28.0317 

0.389121 

27.4522 

0.434449 

26.8080 

35 

0.344849 

32.8455 

0.392202 

32.2041 

0.438727 

31.4878 

40 

0.347153 

37.7208 

o.39559i 

37-0339 

0.443469 

36.2622 

45 

0-34959 

42.6623 

0.39921 

41.9494 

0.44857 

41.1432 

0.80 


0 

0.604367 

o.oooo 

0.635150 

o.oooo 

0.664037 

o.oooo 

s 

0-605352 

3.6792 

0.636270 

3-5256 

0.665292 

3-3709 

10 

0.608313 

7.3783 

0.639647 

7.0709 

0.669078 

6.7608 

15 

0.613278 

11.1174 

0.645321 

10.6563 

0.675454 

10.1903 

20 

0.620284 

14.9189 

0.653363 

14.3047 

0.684523 

13.6808 

25 

0.629382 

18.8064 

0.663856 

18.0400 

0.696420 

17.2583 

30 

0.640619 

22.8058 

0.676907 

21.8900 

0.711313 

20.9506 

35 

0.654029 

26.9467 

0.692614 

25.8869 

0.729386 

24.7922 

40 

0.669614 

31.2614 

0.711054 

30.0675 

0.750820 

28.8239 

45 

0.687309 

35-7856 

0.732244 

34-4731 

0.775752 

33-0928 

Examples,    tanh  (0.75/25°)  =  0.663856/18^.0400. 
tanh  (0.40/20°)  =  0.384135 /i  8°.  1 439^ 


O8] 


TABLE  XIX.    HYPERBOLIC  TANGENTS,    tanh  (p  /6)  =  r  fa.    CONTINUED 


0.15 


0.25 


0.30 


0 

0.148885 

o.oooo 

0.197375 

o.oooo 

0.244919 

0.0000 

0.291312 

0.0000 

5 

0.148901 

4.9261 

0.197414 

4.8697 

0.244992 

4.7986 

0.291435 

4-7133 

IO 

0.148951 

9.8544 

0.197529 

9-7431 

0.245210 

9.6027 

0.291799 

9-4344 

IS 

0.149032 

14.7869 

0.197718 

14.6242 

0.245567 

14.4178 

0.292398 

14.1708 

20 

0.149142 

19.7261 

0.197972 

19-5158 

0.246055 

19.2497 

0.293219 

18.9300 

25 

0.149277 

24.6730 

0.198289 

24.4216 

0.246663 

24.1028 

0.294242 

23-7183 

30 

0.149436 

29.6297 

0.198659 

29-3445 

0.247373 

28.9814 

0.295445 

28.5425 

35 

0.149611 

34-5975 

0.199071 

34.2867 

0.248168 

33.8897 

0.296798 

33-4083 

40 

0.149799 

39-5775 

0.199514 

39-2503 

0.249027 

38.8305 

0.298267 

38-3203 

45 

0.14999 

44.5701 

0.19997 

44.2359 

0.24993 

43-8064 

0.29981 

43.2819 

0.50 

0.55 

0.60 

0.65 

0 

0 

e 

o 

0 

O 

0.462117 

0.0000 

0.500521 

0.0000 

0.537049 

o.oooo 

0.571669 

O.OOOO 

5 

0.462586 

4.2572 

0.501107 

4.1208 

0.537764 

3-9780 

0.572518 

3-8305 

IO 

0.463988 

8.5303 

0.502867 

8.2589 

0.539908 

7-9747 

0.575065 

7-6803 

15 

0.466314 

12.8356 

0.505793 

12.4325 

0.543484 

12.0092 

0.579325 

11.5694 

20 

0.469545 

17.1889 

0.509875 

16.6592 

0.548491 

16.1011 

0.585313 

15.5192 

25 

0.473651 

21.6066 

0.515090 

20-9577 

0.554922 

20.2711 

0.593047 

19-5519 

30 

0.478587 

26.1050 

0.521400 

2S-3472 

0.562757 

24.5408 

0.602535 

23.6917 

35 

0.484289 

30.7008 

0.528746 

29.8475 

0.571953 

28.9336 

0.613763 

27-9645 

40 

0.490664 

35.4089 

0.537034 

34-4778 

0.582428 

334725 

0.626679 

32.3986 

45 

0-49759 

40.2448 

0.546130 

39-2569 

0.594049 

38.1825 

0.641172 

37.0236 

o. 

85 

0.90            0.95 

1.  00 

o 

o 

e 

o 

o 

O 

0.691070 

o.oooo 

0.716298 

O.OOOO 

0.739782 

O.OOOO 

0.761595 

O.OOOO 

5 

0.692454 

3-2158 

0.717804 

3.0617 

0.741402 

2.9094 

0.763317 

2.7603 

10 

0.696634 

6.4500 

0.722358 

6.1406 

0.746304 

5.8339 

0.768533 

5.5300 

15 

0.703692 

9.7217 

0.730063 

9.2539 

0.754611 

8.7900 

0.777391 

8.3322 

20 

0-713763 

13.0522 

0.741095 

12.4228 

0.766541 

11.7964 

0.790146 

11.1764 

25 

0.727041 

16.4666 

0.755702 

15.6708 

0.782407 

14.8755 

0.807174 

14.0855 

30 

0.743766 

19-9939 

0.774215 

19.0269 

0.802627 

18.0556 

0.828989 

17.0858 

35 

0.764227 

23.6700 

0.797038 

22.5275 

0.827737 

21.3722 

0.856264 

20.  2  1  2O 

40 

0.788737 

27-5378 

0.824642 

26.2180 

0.858389 

24.8722 

0.889851 

23-5IOO 

45 

0.817596 

31.6506 

0.857537 

30.1556 

0.895343 

28.6161 

0.930795 

27.O422 

Examples,    tanh  (0.25/25°)  =  o.246663/24°.io28. 
tanh  (0.90/30°)  =  0.774215 1 


1 219  ] 


o 

5 
10 


25 
30 

35 
40 

45 


o 

5 
10 

IS 

20 

25 
30 

35 
40 

45 


o 

5 
10 

IS 

20 

25 
30 

35 
40 

45 


I.OOOOOO 
I.OOOOOO 
I.OOOOOO 
I.OOOOOO 
I.OOOOOO 


o.oooo 
o.oooo 
o.oooo 

0.0000 

o.oooo 


I.OOOOOO  O.OOOO 

I.OOOOOO  0.0000 

I.OOOOOO  O.OOOO 

I.OOOOOO  O.OOOO 

I.OOOOOO  O.OOOO 


0-35 
o 

1.020543  o.oooo 

1.020230  0.2014 

1.019305  0.3969 

1.017796  0.5808 

1.015748  0.7472 

1.013224  0.8908 

1.010302  1.0089 

1.007070  1.0961 

1.003630  1.1503 

1.00008    1.1694 


0.70 

o 

1.083691   o.oooo 
1.082429   0.7875 

1.078682  1.5533 
.072572  2.2755 
.064292  2.9344 

.054105  3-5103 

.042331  3.9861 

.029340  4.3467 

1.015532  4.5800 

1.001334  4-6767 


sinh  0 

TING  FACTOR. 

e   '& 

0.05 

O.IO 

o 

o 

1.000420   o.oooo 

1.001670   o.oooo 

1.000404   0.0042 

1.001638   0.0167 

1.000398   0.0081 

1.001570  0.0325 

1.000373  0.0119 

1.001442   0.0478 

1.000317  0.0153 

1.001279   0.0614 

1.000279   0.0183 

1.001071   0.0730 

1.000216   0.0208 

1.000834   0.0828 

I.OOOI34    O.O222 

1.000571   0.0897 

1.000061   0.0236 

1.000290   0.0942 

i.ooooo    0.0236 

i.ooooo   0.0952 

0.40 

0.45 

o 

o 

.026880   o.oooo 

1.034093   o.oooo 

.026473   0.2625 

1-033577  0.3314 

.025264  0.5172 

1.032044  0.6531 

.023290   0.7569 

1.029542  0.9558 

.020612   0.9742 

1.026148   1.2303 

.017312   1.1625 

1.021969   1.4686 

.013493  1.3161 

1.017131   1.6630 

1.009272   1.4305 

1.011784   I.  8086 

1.004775  1-5017 

1.006092   1.8997 

1.00014   1.5278 

1.00023   1.9338 

o-7S 

0.80 

o 

o 

1.096423   o.oooo 

1.110132   o.oooo 

1.094969   0.9000 

1.108475   1.0192 

1.090658   1.7753 

1.103554  2.0108 

1.083625   2.6019 

1.095532  2.9481 

1.074102   3.3567 

1.084668   3.8044 

1.062383   4-0175 

1.071311  4-5561 

1.048848   4.5647 

1.055881   5.1800 

1.033919   4.9811 

1.038881  5.6567 

1.018059   5-2525 

1.020825   5.9697 

1.001756   5.3678 

1.002274   6.1064 

Example. 


sinh  (0.40/25°) 
0-40/25° 


=  1.017312^ 


C220] 


TABLE  XX.  CORRECTING 

O.IS                   0.20 

FACTO 

D   sinh  0 

=  r£r. 

CONTINUED 
0.30 

e 

0.25 

0 

o 

o 

o 

O 

1-003753 

o.oooo 

1.006680 

o.oooo 

1.010448 

o.oooo 

1.015067 

o.oooo 

5 

1.003696 

0.0372 

1.006579 

0.066  1 

1.010291 

0.1033 

1,014839 

0.1483 

10 

1.003527 

0.0733 

1.006278 

0.1303 

1.009820 

0.2033 

1.014161 

0.2922 

is 

1.003253 

0.0172 

1.005784 

0.1906 

1.009051 

0.2972 

1.013051 

0.4275 

20 

1.002878 

0.1380 

1.005119 

0.2450 

1.008007 

0.3825 

1.011549 

0.5500 

25 

1.002412 

0.1644 

1.004296 

0.2919 

1.006722 

0.4561 

1.000696 

0.6558 

3° 

1.001878 

0.1858 

1.003343 

0.3303 

1.005233 

0.5158 

1.007550 

0.7419 

35 

1.001287 

0.2017 

1.002290 

0.3586 

1.003584 

0.5600 

1.005177 

0.8058 

40 

1.000653 

0.2114 

1.001167 

0.3761 

1.001830 

0.5872 

1.002651 

0.8456 

45 

I.OOOOO 

0.2146 

I.OOOOI 

0.3817 

1.00003 

0.5965 

1.00005 

0.8592 

0.50 

o-SS 

0.60 

0.65 

o 

o 

0 

e 

0 

1.042190 

O.OOOO 

1.051185 

o.oooo 

1.061090 

o.oooo 

1.071920 

o.oooo 

5 

1.041552 

0.4078 

1.050411 

0.4919 

1.060167 

0.5833 

1.070833 

0.6819 

10 

1.039657 

0.8039 

1.048114 

0.9697 

1.057426 

1.1500 

1.0^7611 

1-3477 

IS 

1.036564 

1.1767 

1.044363 

1.4197 

1.052956 

1.6842 

1.062354 

1.9694 

20 

1.032369 

1-5153 

1.039280 

1.8286 

1.046896 

2.1700 

1.055228 

2.5386 

25 

1.027202 

1.8094 

1.033021 

2.1844 

1.039436 

2-5933 

1.046459 

3-0355 

30 

1.021223 

2.0503 

1.025779 

2.4764 

1.030808 

2.9414 

1.036322 

3-4450 

35 

1.014618 

2.2305 

1.017780 

2.6956 

1.021281 

3.2036 

1.025131 

3-7539 

40 

1.007587 

2.3439 

1.009270 

2.8344 

1.011148 

3-370S 

1.013232 

3.9525 

45  • 

1.00035 

2.3872 

1.000508 

2.8880 

1.000720 

3-4369 

1.000991 

4-0330 

0.85 

0.00 

0.95 

1.  00 

o 

o 

o 

o 

0 

1.124842 

o.oooo 

1.140574 

o.oooo 

1-157352 

o.oooo 

1.175201 

O.OOOO 

5 

1.122964 

1.1447 

1.138461 

1.2767 

1.154989 

1.4147 

1-172573 

1-5592 

10 

I.II7393 

2.2592 

1.132194 

2.5200 

1.147982 

2.7928 

1.164779 

3-0775 

IS 

1.108311 

3-3131 

1.121981 

3.6967 

1.136564 

4.0986 

1.152083 

4.5180 

20 

1.096014 

4.2775 

I.  108160 

4-7753 

I.I2III7 

5.2969 

1.134913 

5-8417 

25 

1.080904 

5-1255 

1.091177 

5-7253 

1.102151 

6.3544 

1.113841 

7.0122 

3° 

1.063461 

5.8314 

1.071587 

6.5183 

1.080280 

7.2400 

1.089558 

7-9955 

35 

1.044241 

6.3730 

1.050012 

7.1294 

1.056208 

7.9253 

1.062847 

8.7603 

40 

1.023843 

6.7314 

1.027127 

7-5372 

1.030691 

8.3864 

1.034550 

9.2789 

45 

1.002897 

6.8917 

1.003640 

7.7242 

1.004518 

8.6033 

1.005545 

9.5292 

Example. 


sinh  (0.95/25°)  _ 
0-95/25^ 


[221] 


TABLE  XXI.    CORRECTING  FACTOR. 


taphg 

e 


0.05 


0 

1.  000000 

o.oooo 

0.999160 

o.oooo 

0.996680 

o.oooo 

S 

I.OOOOOO 

o.oooo 

0.999177 

0.0083 

0.996732 

0.0330 

10 

I.OOOOOO 

o.oooo 

0.999223 

0.0163 

0.996882 

0.0653 

IS 

I.OOOOOO 

o.oooo 

0.999291 

0.0239 

0.997119 

0.0950 

20 

I.OOOOOO 

o.oooo 

0.999359 

0.0308 

0.997448 

0.1222 

25 

I.OOOOOO 

o.oooo 

0.999475 

0.0364 

0.997858 

0.1459 

3° 

I.OOOOOO 

o.oooo 

0.999591 

0.0411 

0.998331 

0.1650 

35 

I.OOOOOO 

o.oooo 

0.999707 

0.0450 

0.998856 

0.1792 

40 

I.OOOOOO 

o.oooo 

0.999844 

0.0469 

0.999414 

0.1878 

45 

I.OOOOOO 

o.oooo 

I.  00000 

0.0480 

0.99999 

0.1913 

0.35 


0.40 


0.45 


o 

0.961071 

o.oooo 

0.949872 

o.oooo 

0.937553 

o.oooo 

5 

0.961606 

0.3847 

0.950538 

0.4942 

0-938355 

0.6139 

10 

0.963197 

0.7592 

0.952528 

0.9761 

0.940753 

1.2138 

i5 

0.965817 

1.1142 

0.955812 

1.4342 

0.944719 

1.7853 

20 

0.969419 

1.4400 

0.960338 

1.8561 

0.950209 

2.3147 

25 

0.973929 

1.7281 

0.966036 

2.2311 

0.957150 

2.7881 

3° 

0.979258 

1.9683 

0.972801 

2.5478 

0.965442 

3.1920 

35 

0.985284 

2.1545 

0.980505 

2.7959 

0.974949 

3.5122 

40 

0.991867 

2.2792 

0.988977 

2.9661 

0.985487 

3-7378 

45 

0.99883 

2.3377 

0.99800 

3.0506 

0.99684 

3.8568. 

0.70 


0.75 


0.80 


o 

0.863381 

o.oooo 

0.846867 

o.oooo 

0.830046 

o.oooo 

5 

0.864789 

1.3208 

0.848360 

1-4744 

0.831615 

1.6291 

10 

0.869019 

2.6217 

0.852862 

2.9291 

0.836347 

3.2392 

i5 

0.876111 

3.8826 

0.860428 

4-3437 

0.844318 

4.8097 

20 

0.886121 

5.0811 

0.871150 

5-6953 

0-855654 

6.3192 

25 

0.899117 

6.1936 

0.885142 

6.9600 

0.870525 

7-7417 

30 

0.915169 

7.1942 

0.902542 

8.1100 

0.889141 

9.0494 

35 

0.934328 

8.0533 

0.923485 

9.1131 

0.911732 

10.2078 

40 

0.956592 

8.7386 

0.948072 

9.9325 

0.938525 

11.1761 

45 

0.981870 

9.2144 

0.976325 

10.5269 

0.969690 

11.9072 

Note.  Negative  quantities  are  in  heavy  type. 


Example. 


tanh  (0.75/25°) 
0.75/25° 


[223] 


XAXAJV^  ±  O.J.TI  VJ   1  f\\-,  J.  V 

e   rti- 

^UNTLNUED 

o.i  5 

O.2O 

0.25 

0.30 

0 

o 

0 

0 

o 

O 

0.992567  o.oooo 

0.986875  o.oooo 

0.979676  o.oooo 

0.971040  o.oooo 

5 

0.992676  0.0739 

0.987069  0.1303 

0.979968  0.2014 

0.971449  0.2867 

10 

0.993006  0.1456 

0.987642  0.2569 

0.980839  0.3973 

0.972665  0.5656 

15 

0.993547  0.2131 

0.988588  0.3758 

0.982269  0.5822 

0.974660  0.8292 

20 

0.994280  0.2739 

0.989860  0.4842 

0.984222  0.7503 

0-977396  1.0700 

25 

0.995182  0.3270 

0.991446  0.5784 

0.986650  0.8972 

0.980807  1.2817 

30 

0.996237  0.3703 

0.993295  0.6555 

0.989493  1.0186 

0.984817  1.4575 

35 

0.997410  0.4025 

0.995357  0.7133 

0.992673  x.i  103 

0.989327  1.5917 

40 

0.998660  0.4225 

0.997571  0.7497 

0.996106  1.1695 

0.994224  1.6797 

45 

0.99995  0.4299 

0.99987   0.7641 

0.99971   1.1936 

0.99937  1-7181 

0.50 

0.55 

O.6o 

0.65 

0 

0 

o 

o 

o 

o 

0.924234  o.oooo 

0.910038  o.oooo 

0.895082  o.oooo 

0.879491  o.oooo 

5 

0.925172  0.7428 

0.91  1  104  0.8792 

0.896274   I.O22O 

0.880796  1.1695 

10 

0.927976  1.4697 

0.914304  1.7411 

0.899847   2.0253 

0.884716  2.3197 

15 

0.932627  2.1644 

0.919624  2.5675 

0.905806   2.9908 

0.891269  3.4306 

20 

0.939089  2.8111 

0.927046  3.3408 

0.914151   3.8989 

0.900482  4.4808 

25 

0.947302  3-3934 

0.936527  4-0423 

0.924870  4.2289 

0.912380  5.4481 

30 

0.957174  3-8950 

0.948001  4.6528 

0.937928   5-4592 

0.926976  6.3083 

35 

0.968577  4.2992 

0.961356  5-1525 

0.953254   6.0664 

0.944250  7-0355 

40 

0.981327  4.59" 

0.976426  5.5222 

0.970713   6.5275 

0.964122  7.6014 

45 

0.99516   4-7552 

0.992963  5.7431 

0.990081   6.8175 

0.986419  7.9764 

0.85 

0.90 

0.95 

1.00 

o 

o 

0 

0 

o 

O 

0.813024  o.oooo 

0.795887  o.oooo 

0.778718  o.oooo 

0.761595  o.oooo 

5 

0.814651  1.7842 

0.797560  1.9383 

0.780424  2.0906 

0.763317  2.2397 

IO 

0.819569  3-5500 

0.802620  3.8594 

0-785583  4-1661 

0-768533  4.4700 

15 

0.827873  5-2783 

0.811181  5.7461 

0.794327  6.2100 

0.777391  6.6678 

20 

0.839722  6.9478 

0.823438  7-5772 

0.806886  8.2036 

0.790146  8.8236 

25 

0.855342  8.5334 

0.839669  9.3292 

0.823586  10.1245 

0.807174  10.9145 

30 

0.875019  10.0061 

0.860239  10.9731 

0.844870  11.9444 

0.828989  12.9142 

35 

0.899091  11.3300 

0.885598  12.4725 

0.871302  13.6278 

0.856264  14.7880 

40 

0.927926  12.4622 

0.916269  13.7820 

0.903567  15.1278 

0.889851  16.4900 

45 

0.961877  13.3494 

0.952819  14-8444 

0.942466  16.3839 

0.930795  17.9578 

Note.          Negative  quantities  are  in  heavy  type. 

tanh  (1.0/10°) 
Eiample.    : — 5 =  0.768533X4^4700. 


[223] 


TABLE  XXIL    FUNCTIONS  OF  SEMI-IMAGINARIES 
COMPLEX  VARIABLE  0/45°  (slope  constant). 


9 

Sinh0/45° 

Cosh  0/45° 

tanh  6  /45° 

Coseche/45° 

hyp. 

Size 

Slope 

Size  ^ 

Slope 

Size  _ 

Slope 

Size 

Slope 

rads. 

numeric 

degrees 

numeric 

degrees 

numeric 

degrees 

numeric  degrees 

0.00 

o.ooooo 

'45.0000 

I.OOOOO 

o.oooo 

o.ooooo 

45.0000 

oc 

45.0000 

O.OI 

O.OIOOO 

45.0009 

I.OOOOO 

0.0028 

O.OIOOO 

44.9981 

IOO.OOOOO 

45.0009 

O.O2 

0.02OOO 

45-0037 

I.OOOOO 

0.0114 

O.O2OOO 

44.9923 

50.00000 

45-0037 

0.03 

0.03000 

45.0084 

I.OOOOO 

0.0257 

0.03000 

44.9827 

33-33333 

45.0084 

0.04 

0.04000 

4S-oi5i 

I.OOOOO 

0.0458 

0.04000 

44-9693 

25.00000 

45.0151 

O.OS 

0.05000 

45-o236 

I.OOOOO 

0.0716 

0.05000 

44.9520 

20.00000 

45-0236 

O.O6 

0.06000 

45-0342 

I.OOOOO 

0.1031 

0.06000 

44.9310 

16.66667 

45-0342 

O.O? 

0.07000 

45-0465 

.00000 

0.1403 

0.07000 

44.9062 

14.28571 

45-0465 

0.08 

0.08000 

45.0608 

.00000 

0.1833 

0.08000 

44-8775 

12.50000 

45.0608 

O.OQ 

0.09000 

45.0770 

.00001 

0.2320 

O.OQOOO 

44.8450 

n.  mil 

45-0770 

O.IO 

O.IOOOO 

45-0952 

.00001 

0.2864 

O.IOOOO 

44.8087 

10.00000 

45-0952 

O.II 

O.IIOOO 

45-"52 

.00001 

0.3466 

O.IIOOO 

44.7686 

9.09091 

45.«52 

O.I2 

O.I2OOO 

45-I372 

.00002 

0.4125 

O.I2OOO 

44.7247 

8-33333 

45.1372 

0.13 

0.13000 

45.1611 

1.00003 

0.4841 

0.13000 

44.6770 

7-69231 

45.1611 

0.14 

0.14000 

45.1869 

1.00004 

0.5614 

0.13999 

44-6255 

7.14286 

45-1869 

0.15 

0.15000 

45.2146 

1.00005 

0.6445 

0.14999 

44-5701 

6.66667 

452146 

0.16 

0.16000 

45.2442 

1.00006 

0-7333 

0.15999 

44.5109 

6.25000 

45.2442 

0.17 

0.17000 

45-2757 

1.00008 

0.8278 

0.16999 

44-4479 

5-88235 

45-2757 

0.18 

0.18000 

45-3092 

1.00009 

0.9281 

0.17998 

44.3811 

5.55556 

453092 

0.19 

0.19000 

45-3445 

I.OOOII 

1.0341 

0.18998 

44.3104 

5-26316 

45-3445 

O.20 

O.20OOO 

45-38I7 

1.00013 

1.1458 

0.19997 

44-2359 

5.00000 

45-38I7 

O.2I 

O.2IOOO 

45.4208 

1.  00016 

1.2632 

0.20997 

44.1576 

4.76190 

454208 

0.22 

O.22OOO 

45.4619 

I.OOO2O 

1.3864 

0.21996 

44-0755 

4-54545 

45.4619 

0.23 

0.23000 

45-5048 

1.00024 

1-5152 

0.22995 

43.9896 

4.34783 

45.5048 

O.24 

0.24000 

45-5497 

1.00028 

1.6498 

0.23994 

43.8999 

4.16667 

45-5497 

0.25 

0.25001 

45-5965 

1.00033 

1.7901 

0.24993 

43.8064 

3.99984 

45.5965 

O.26 

0.26001 

45-6453 

1.00038 

1.9361 

0.25991 

43.7092 

3.84601 

45-6453 

0.27 

0.27001 

45-6959 

1.00044 

2.0878 

0.26989 

43.6081 

3.70357 

45-6959 

0.28 

0.28001 

45-7485 

1.00051 

2.2452 

0.27987 

43-5032 

3-57I30 

45.7485 

O.29 

0.29001 

45.8029 

1.00060 

2.4084 

0.28984 

43-3945 

3.44816 

45.8029 

0.30 

0.30001 

45-8592 

1.00068 

2-5773 

0.29981 

43.2819 

3-33322 

45-8592 

0.31 

0.31002 

45-9I74 

1.00077 

2.7520 

0.30978 

43-1654 

3.22560 

45.9174 

0.32 

0.32002 

45-9775 

1.00087 

2.9323 

0.31974 

43-0452 

3.12481 

45-9775 

0-33 

0.33002 

46.0396 

1.00098 

3.«83 

0.32970 

42-9213 

3.03012 

46.0396 

0-34 

0.34002 

46.1036 

I.OOIII 

3-3099 

0.33965 

42-7937 

2.94100 

46.1036 

0-35 

0.35003 

46.1694 

1.00125 

3-507I 

0-34959 

42.6623 

2.85690 

46.1694 

0.36 

0.36003 

46.2372 

1.00140 

3-7ioo 

0-35953 

42.5272 

2-77755 

46.2372 

0-37 

0.37004 

46.3069 

1.00156 

3-9i85 

0.36946 

42.3884 

2.70241 

46.3069 

0.38 

0.38004 

46.3786 

1.00174 

4-1328 

0-37939 

42-2458 

2.63130 

46.3786 

0-39 

0.39005 

46.4522 

1.00193 

4-3528 

0.38930 

42.0994 

2.56377 

46.4522 

0.40 

0.40006 

46-5278 

1.00213 

4-5784 

0.39921 

41.9494 

2.49963 

46.5278 

0.41 

0.41006 

46.6053 

1.00235 

4.8096 

0.40910 

41-7957 

2.43867 

46.6053 

0.42 

0.42007 

46.6846 

1.00259 

5-0464 

0.41899 

41.6382 

2.38056 

46.6846 

0.43 

0.43008 

46-7658 

1.00285 

5.2889 

0.42886 

41.4778 

2-32515 

46.7658 

o-44 

0.44009 

46.8489 

1.00312 

5-5369 

0.43872 

41.3120 

2.27226 

46.8489 

0-45 

0.45010 

46.9338 

1.00341 

5.7906 

0.44857 

41.1432 

2.22173 

46.9338 

0.46 

0.4601  2 

47.0206 

1.00372 

6.0498 

0.45841 

40.9708 

2-17335 

47.0206 

0.47 

0.47013 

47.1093 

1.00406 

6.3146 

0.46823 

40.7947 

2.12707 

47.1093 

0.48 

0.48014 

47.2000 

1.00442 

6.5850 

0.47803 

40.6150 

2.08273 

47.2000 

0.49 

0.49016 

47.2926 

1.00480 

6.8609 

0.48782 

40.4317 

2.04015 

47.2926 

0.50 

0.50017 

47-3872 

1.00520 

7.1424 

0-49759 

40.2448 

1.99932 

47.3873 

Note. 


Negative  quantities  are  in  heavy  type. 


Examples,    sinh  0.13  745°  =*  0.13000  745°.  1611. 
cosh 0.26/45°  =  1.00038  /i.°936i. 

[224] 


TABLE  XXII.    FUNCTIONS  OF  SEMI-IMAGINARIES.    CONTINUED 
COMPLEX  VARIABLE  6  /45°  (slope  constant). 


e 

Sech  6  /45° 

Coth  0  /45° 

Sinh  0  /45°  /  9  /45° 

tanh  B  /4S°  /  0  /45° 

hyp. 

Size_ 

Slope 

Size 

Slope 

Size 

Slope 

Size 

Slope 

rads. 

numeric 

degrees 

numeric 

degrees 

numeric 

degrees 

numeric 

degrees 

0.00 

I.OOOOO 

o.oooo 

oc 

45.0000 

I.OOOOO 

o.oooo 

I.OOOOO 

o.oooo 

0.01 

I.OOOOO 

0.0028 

IOO.OOOOO 

44.9981 

I.OOOOO 

0.0009 

I.OOOOO 

0.0019 

O.02 

I.OOOOO 

0.0114 

50.00000 

449923 

I.OOOOO 

0.0037 

I.OOOOO 

0.0077 

0.03 

I.OOOOO 

0.0257 

33-33333 

44.9827 

I.OOOOO 

0.0084 

I.OOOOO 

0.0173 

O.O4 

I.OOOOO 

0.0458 

25.00000 

449693 

I.OOOOO 

0.0151 

I.OOOOO 

0.0307 

0.05 

I.OOOOO 

0.0716 

20.00000 

449520 

I.OOOOO 

0.0236 

I.OOOOO 

0.0480 

O.O6 

I.OOOOO 

0.1031 

16.66667 

44.9310 

I.OOOOO 

0.0342 

I.OOOOO 

0.0690 

O.O7 

I.OOOOO 

0.1403 

14.28571 

44.9062 

I.OOOOO 

0.0465 

0.99999 

0.0938 

0.08 

I.OOOOO 

0.1833 

12.50000 

44-8775 

I.OOOOO 

0.0608 

0-99999 

0.1225 

O.OQ 

0.99999 

0.2320 

ii.iiin 

44.8450 

I.OOOOO 

0.0770 

0.99999 

0.1550 

O.IO 

0.99999 

0.2864 

IO.OOOOO 

44-8087 

I.OOOOO 

0.0952 

0-99999 

0.1913 

O.I  I 

0.99999 

0.3466 

9.09091 

44.7686 

I.OOOOO 

0.1152 

0.99998 

0.2314 

0.12 

0.99998 

0.4125 

8-33333 

44-7247 

I.OOOOO 

0.1372 

0.99998 

0.2753 

0.13 

0.99997 

0.4841 

7.69231 

44.6770 

I.OOOOO 

0.1611 

0.99997 

0.3230 

O.I4 

0.99996 

0.5614 

7-14337 

446255 

I.OOOOO 

0.1869 

0.99996 

0-3745 

0.15 

0.99995 

0.6445 

6.66712 

445701 

I.OOOOO 

0.2146 

099995 

0.4299 

O.l6 

0.99994 

0-7333 

6.25039 

445109 

I.OOOOO 

0.2442 

0.99994 

o  4891 

O.I? 

0.99992 

0.8278 

5.88271 

444479 

I.OOOOI 

0.2757 

0.99992 

0.5521 

0.18 

0.99991 

0.9281 

S-SS6i8 

44.3811 

I.OOOOI 

0.3092 

0.99991 

06189 

O.I9 

0.99989 

•0341 

5.26371 

44.3104 

I.OOOOI 

0-3445 

0.99989 

0.6896 

O.2O 

0.99987 

.1458 

5.00050 

44-2359 

I.OOOOI 

0.3817 

0.99987 

0.7641 

0.21 

0.99984 

.2632 

4-76258 

441576 

I.OOOO2 

0.4208 

0.99984 

0.8424 

O.22 

0.99980 

.3864 

4.54628 

44-0755 

I.OOOO2 

0.4619 

0.99981 

0.9245 

0.23 

0.99976 

.5152 

4-34878 

43.9896 

I.OOOO2 

0.5048 

0.99978 

1.0104 

O.24 

0.99972 

1.6498 

4.16771 

438999 

I.OOOO2 

0-5497 

0-99975 

I.IOOI 

0.25 

0.99967 

1.7901 

4.OOI  I  2 

43.8064 

1.00003 

0-5965 

0.99971 

1.1936 

0.26 

0.99962 

1.9361 

3.84748 

43.7092 

1.00003 

0-6453 

0.99966 

2.2908 

0.27 

0.99956 

2.0878 

3.70521 

43.6081 

1.00003 

0-6959 

0.99960 

1.3919 

0.28 

0.99949 

2.2452 

3-57309 

43.5032 

1.00004 

07485 

0-99953 

1.4968 

O.29 

0.99940 

2.4084 

3.45018 

433945 

1.00004 

0.8029 

0-99945 

1.6055 

0.30 

0.99932 

2-5773 

3-33544 

43  2819 

1.00005 

0.8592 

0-99937 

1.7181 

0.31 

0.99923 

2.7520 

3.22810 

43.1654 

1.00005 

0.9174 

0.99928 

1.8346 

0.32 

0.99913 

29323 

3-I2754 

430452 

1.00006 

0-9775 

0.99919 

1.9548 

0-33 

0.99902 

3.1183 

3-0330S 

42.9213 

1.00006 

1.0396 

0.99908 

2.0787 

0-34 

0.99889 

3  3099 

2.94421 

42-7937 

1.00007 

1.1036 

0.99896 

2.2063 

0-35 

0.99875 

3-5071 

2.86049 

42.6623 

1.00008 

1.1694 

0.99883 

2.3377 

0.36 

0.99860 

3.7100 

2.78141 

42.5272 

1.00009 

1.2372 

0.99869 

2.4728 

o-37 

0.99844 

39185 

2.70665 

42-3884 

I.OOOIO 

1.3069 

0.99854 

2.6116 

0.38 

0.99826 

4-1328 

2.63580 

42.2458 

I.OOOII 

1-3786 

0-99837 

3.7543 

o-39 

0.99807 

43528 

2.56870 

42.0994 

1.00013 

1.4522 

0.09819 

2.9006 

0.40 

0.99788 

4-5784 

2.50494 

42.9494 

1.00014 

1.5278 

0.99800 

3.0506 

0.41 

0.99766 

4.8096 

2.44438 

4i  7957 

1.00016 

1.6053 

0.99780 

3.2043 

0.42 

0.99742 

5.0464 

2.38668 

41.6382 

1.00017 

1.6846 

0-99759 

3.3618 

o-43 

0.99716 

5.2889 

2.33174 

41.4778 

1.00019 

1.7658 

0.99736 

3.5230 

o-44 

0.99689 

5.5369 

2-27935 

41.3120 

I.OOO2I 

1.8489 

0.99711 

3.6880 

0-45 

0.99660 

5.7906 

2.22930 

41  US* 

1.00023 

I.9338 

0.99684 

3.8568 

0.46 

0.99629 

6.0498 

2.18145 

40.9708 

1.00025 

2.0206 

0.99655 

4.0292 

0.47 

0.99596 

6.3146 

2.I357I 

40.7947 

1.00027 

2.1093 

0.99624 

4.2053 

0.48 

0.99560 

6.5850 

2.09190 

40.6150 

1.00030 

2.2OOO 

0.99591 

4.3850 

0-49 

0.99522 

6.8609 

2.04995 

40.4317 

1.00032 

2.2926 

0-99555 

4.5683 

0.50 

0^9483 

7.1424 

2.00969 

40.2448 

1.00035 

2.3872 

0.99516 

4-7552 

Note. 

Negative  quantities  are  in  heavy  type. 

Examples. 


tanh  0.39  /45° 

o-39 
coth  0.50  /45° 


TABLE  XXIII.    HYPERBOLIC  FUNCTION  FORMULAS 

(from  Smithsonian  Mathematical  Tables  No.  1871  of  1909,  Becker  and  van  Orstrand's 
"Hyperbolic  Functions,"  by  permission.) 

A.    RELATIONS  BETWEEN  HYPERBOLIC  AND  CIRCULAR  FUNCTIONS 

1.  sinh  u  =  —  i  sin  iu  =  tan  gd  u. 

2.  cosh  u  =  cos  iu  =  sec  gd  u. 

3.  tanh  u  =  —  i  tan  iu  =  sin  gd  u. 

4.  tanh  \u—  tan  \  gd  u. 

5.  0"  =  (i  +  sin  gd  u)  -i-  cos  gd  u, 

=  [i  -  cos  (J  TT  +  gd  tt)3  -i-  sin  (£  TT  +  gd  «), 
=  tan  (JTT  +  %  gdu). 

6.  sinh  iu  =  i  sin  u. 

7.  cosh  ra  =  cos  u. 

8.  tanh  *«  =  i  tan  w. 

8a.  sin  u  =  —  i  sinh  iu  =  tanh  (grf"1  u). 
8b.  cos  M  =  cosh  iu  =  sech  (gd""1  u). 
8c.  tan  M  =  —  i  tanh  i«  =  sinh  (gd~l  u). 

9.  sinh  (u  ±  w>)  =  ±  i  sin  (z>  =F  i«), 

=  sinh  u  cos  v  ±  i  cosh  w  sin  v. 

10.  cosh  (w  ±  iv)  =  cos  (v  T  iu), 

=  cosh  M  cos  v  db  t  sinh  w  sin  w. 

loa.  sin  (w  =t  *»)  =  ±  i  sinh  (t»  ±  tw)  =  sin  «  cosh  v  ±  i  cos  «  sinh  v. 
xob.  cos  («  =b  iv)  =  cosh  (v  =F  iu)  =  cos  M  cosh  v  =F  i  sin  «  sinh  ». 

1 1 .  cosh  (»w  TT)  =  cos  m  IT.     (m  is  an  integer.) 

12.  sinh  (2  w  +  i)  \  i  IT  =  i  sin  (2  w  +  i)  I  T.     (m  is  an  integer.) 

B.    RELATIONS  AMONG  THE  HYPERBOLIC  FUNCTIONS 

13.  sinh  u=  \(eu  —  e~u)  =  —  sinh  (—  u)  =  (csch  u)  ~l, 

=  2  tanh  \  u  -i-  (i  —  tanh2  \  u)  =  tanh  u  -f-  (i  —  tanh  2  «)*. 

14.  cosh  «  =  £  (eu  +  e~")  =  cosh  (—  u)  =  (sech  u)~l, 

=  (i  +  tanh2  i  u)  -5-  (i  -  tanh2 }  M)  =  i  -=-  (i  -  tanh2«)*. 

15.  tanh  u  =  (eu  -  e~")  -T-  (eu  +  «r«)  =  -  tanh  (-  w), 

=  (coth  u)~l  =  sinh  M  -i-  cosh  M  =  (i  —  sech2  M)*. 

16.  sechtt  =  sech  (—  w)  =  (i  —  tanh2«)*. 

[226] 


FORMULAS 

17.  cschw  =  —  csch  (—  «)  =  (coth2w  —  i)*. 

1 8.  coth  u  =  -  coth  (-  u)  =  (csch2  w  +  i)*. 

19.  cosh2  u  —  sinh2  u  =  i. 


20.     sinh  £  «  =  V  i  (cosh  w  —  i). 


21.  cosh  %  w  =   V  i  (cosh  w  +  i). 

22.  tanh  ^  «  =  (cosh  w  —  i)  -5-  sinh  u,        

=  sinh  u  -T-  (i  +  cosh  w)  =  V  (cosh  u  —  i)  -r-  (cosh  M  +  i). 

23.  sinh  2«  =  2  sinh  w  cosh  u  =  2  tanh  M  -f-  (i  —  tanh2  u). 

24.  cosh  2«  =  cosh2  u  +  sinh2  w  =  2  cosh2  M  —  i, 

=  1+2  sinh2«  =  (i  +  tanh2«)  -f-  (i  —  tanh2w). 

25.  tanh  2U  =  2  tanh  M  -f-  (i  +  tanh2«). 

26.  sinh  3«  =  3  sinh  w  +  4  sinh3 «. 

27.  cosh  3«  =  4  cosh3 «  —  3  cosh  u. 

28.  tanh  3«  =  (3  tanh  u  +  tanh3  w)  -f-  (i  +  3  tanh2 «). 
28a>  m  cosh  u  +  n  sinh  w  =  J  (w  +  »)  eu  +  \  (m  —  ri)  e~". 
a8b.  m  eu  ±  w«e~u  =  (m  ±  ri)  cosh  w  +  (w  =F  w)  sinh  u. 

29.  sinh  «M  = 

n  cosh""1  u  sinh  w  +  ^-^ -4-J cosh""3  M  sinh3 «+.... 

6 

/M     ('M     — —      T  ) 

30.  cosh  nu  =  cosh"  u  -\ —  —  cosh  n~2  u  sinh2  u  + 

2 

31.  sinh  w  +  sinh  i>  =  2  sinh  \  (u  -\-  v)  cosh  §  (M  —  z>). 

32.  sinh  w  —  sinh  v  =  2  cosh  !(«  +  »)  sinh  %  (u  —  v). 

33.  cosh  u  -f  cosh  0  =  2  cosh  £  («  +  v)  cosh  £  («  —  v). 

34.  cosh  «  —  cosh  v  =  2  sinh  %(u  +  v)  sinh  HM  ~  *)• 

35.  sinh  u  +  cosh  M  =  (i  +  tanh  £  w)  -H  (i  —  tanh  i  w). 

36.  (sinh  w  +  cosh  u)n  =  cosh  w«  +  sinh  nu. 

36a.  a  sinh  u  +  b  cosh  u  =  V#2  ~  ^2  sinh  («  +  tanh"1  ^).  a>  b 

=  V*2-  a2  cosh  («  +  coth"1 1).  6  >  a 

__^____^^  i 

36b.  a  cosh  «  ±  i  sinh  u  =  Va2  —  62  cosh  (w  ±  tanh"1  j). 

37.  tanh  w  +  tanh  z>  =  sinh  (M  +  v)  •+•  cosh  w  cosh  v. 

38.  tanh  M  —  tanh  v  =  sinh  (u  —  v)  -f-  cosh  w  cosh  ». 

39.  coth  u  +  coth  t>  =  sinh  («  +  v)  -i-  sinh  M  sinh  ». 

[227] 


FORMULAS 

40.  coth  u  —  coth  v  =  —  sinh  (u  —  v)  -f-  sinh  u  sinh  v. 

41.  sinh  (u  ±  v)  =  sinh  w  cosh  z>  =t  cosh  M  sinh  t>. 

42.  cosh  (u  =t  z>)  =  cosh  w  cosh  z;  ±  sinh  M  sinh  z>. 

43.  tanh  (u  ±  z»)  =  (tanh  w  ±  tanh  v)  -H  (i  ±  tanh  w  tanh  z>). 

44.  coth  (u±  v)  =  (coth  M  coth  z>  ±  i)  -f-  (coth  z>  ±  coth  M). 

45.  sinh  (u  +  v)  +  sinh  (w  —  zi)  =  2  sinh  u  cosh  z>. 

46.  sinh  (u  +  z>)  —  sinh  («  —  z>)  =  2  cosh  w  sinh  v. 

47.  cosh  («  +  »)+  cosh  (u  —  v)  =  2  cosh  u  cosh  z>. 

48.  cosh  (u  +  v)  —  cosh  (M  —  »)  =  2  sinh  u  sinh  v. 

49.  tanh  \  (u  +  z;)  =  (sinh  M  -f  sinh  z>)  -f-  (cosh  M  +  cosh  v). 

50.  tanh  ^  (w  —  v)  =  (sinh  M  —  sinh  v)  -f-  (cosh  w  +  cosh  z>). 

51.  coth  J  (M  +  z>)  =  (sinh  w  —  sinh  v)  -t-  (cosh  M  —  cosh  »). 

52.  coth  5  («  —  »)  =  (sinh  M  +  sinh  z>)  -f-  (cosh  «  —  cosh  z>). 

tanh  «  +  tanh  v  _  sinh  (w  +  v) 
tanh  M  —  tanh  v       sinh  («  —  z;)  * 

coth  u  +  coth  v  _  _  sinh  (w  +  ») 
coth  M  —  coth  z»          sinh  (w  —  v) 

55.  sinh  (w  +  z>)  +  cosh  («  +  v)  =  (cosh  w  +  sinh  M)  (cosh  »  +  shih  v). 

56.  shih  (w  +  v)  sinh  («  —  »)  =  sinh2  u  —  sinh2 », 

=  cosh2  u  —  cosh2  z>. 

57.  cosh  (u  +  z>)  cosh  (u  —  v)  =  cosh2  u  +  sinh2 », 

=  sinh2  M  +  cosh2  v. 

58.  sinh  (#w  TT)  =  o.     (m  is  an  integer.) 

59.  cosh  (miir)  =  (  —  i)m. 

60.  tanh  (mi  TT)  =  o. 

61.  sinh  (u  +  miir)  =  (—  i)msinh«. 

62.  cosh  (u  +  mi  IT)  =  (—  i)mcoshw. 

63.  sinh  (2  m  +  i)  £  *'  TT  =  ±  i. 

64.  cosh  (2  w  +  i)  3  i  TT  =  o. 

65.  sinh  ( —  ±  « )  =  i  cosh  M. 

\  2  / 


.    cosh  ( —  ±w)  =  =t  *  sinh  M. 
\2         / 


66.  

[228] 


FORMULAS 

66a.  sinh  {  (u  +  iv)  +  i  f  \  =  sinh  {  (u  +  iv)  +  i  i  \  —  i  cosh  (u  +  iv). 
66b.  cosh  \  (u  +  iv)  +  i  f  |  =  cosh  |  (w  +  «>)  +  «  T;  \  =  i.sinh  (M  +  iv). 
66c.  tanh  ^  (w  +  iv)  +  i  f  |  =  tanh  |  (w  +  iv)  +  i  i_  \  =  coth  (u  +  iv). 
66d.  sinh  |  (w  +  iv)  +  i TT  \  =  sinh  \  (u  +  iv)  +  is_\  =  —sinh  (u  +  iv). 
66e.  cosh  \  (u  +  iv)  +  i  *  \  =  cosh  \  (u  +  iv)  +  i  2  \  =  —cosh  (u  +  iv). 
66f.   tanh  \  (u  +  iv)  +  i  T  \  =  tanh  \  (u  +  iv)  +  i  2  \  =  tanh  (u  +  iv). 
67.    tanh  (u  +  iir)  =  tanh  u. 

67a.  If  sinh  ^  (u  +  t  (i  -  q)  \  =  x  +  iy;  then  sinh  \  u  +  i  (i  +  q)  \  =-  x  +  iy. 
670.  If  cosh  \(u  +  i(i-q)\=x  +  iy:  then  cosh  \  u  +  i  (i  +  q)  \  =-  x  +  iy. 
67C.  If  tanh  \  u  +  i  (i  -  q)  \  =  x  +  iy:  then  tanh  \  u  +  i  (i  4-  q)  I  =  x  -  iy. 

C.    INVERSE  HYPERBOLIC  FUNCTIONS 

/T 
T-TI — u  * 
^  +  i;» 

69.  cosh-1  u  =  log  (w  +  Vw2  —  i)  =  sinh  -1  V«2  -  i  =   /  ,  2  _    ^  • 

r  du 

70.  tanh"1  w  =  \  log  (i  +  «)  —  £  log  (i  —  u)  =  I  2* 

%/       A  ** 

71.  coth"1  M  =  ^  log  (i  +  u)  —  i  log  (u  —  i)  =  /  j  _  W2  : 


72. 

73.  csch"1  u  = 

74.  sin"1  w  =  —  «  sinh"1  iu  =—  i  log  (I'M  +  V1  —  «2)- 

75.  cos"1  u  =—  i  cosh"1  M  =  —  i  log  (M  +  i  Vi  ~  «2)- 

76.  tan-1  u=—  i  tanh"1  ZM  =  — : log  (i  +  iu) .  log  (i  —  iu). 

•  t      1.  11: 


21 


77.  cot'1  u-  i  coth-1  iu  =  — .  log  (iu  —  i)  —  TJ  J°g  (*M  +  J)- 

2  Z*  •  ** 

78.  sin"1  iu  =  i  sinh"1  u  =  i  log  (w 


C  229 .1 


FORMULAS 

1  7 

80.  tan"1  iu  =  i  tanh-1  u  =  -  log  (i  +  «)  —  -  log  (i  —  «). 

2  2 

81.  cot-1  iu  =—  i  cotir1  u  =  --  log  (w  +  i)  +  —  log  (u  —  i). 

2  2 

82.  cosh-1  \(u  +  -}  =  sinh-1  \(u  -  -}  = 

\  «/  \  «/ 


tt 


i-    i        ~ 

=  2  tanh-1         -  =  log  M. 
«  +  i 

83.  tanh-1  tan  M  =  \gdzu. 

84.  tan"1  tanh  M  =  \  gd~l  2  «. 

85.  cosh"1  esc  2  «  =  —  sinh-1  cot  2  u  =  —  tanh-1  cos  2  u  =  log  tan  u. 

86.  tanh-1  tan2  (J  TT  +  i  «)  =  J  log  esc  w. 

87.  tankr1  tan2  1  «  =  ^  log  sec  u. 

88.  cosh"1  «  ±  cosh"1  v  =  cosh"1  [«»  ±  V  (w2  —  i)  (v2  —  i)]. 

89.  sinh"1  «  ±  sinh"1  z>  =  sinh-1  \ju  V  i  +  v2  ±  z;  V  i  +  w2]. 

D.    SERIES 

M2         7<3         M4 

90.  e«=I+M  +  -  +  -  +  -+  .  ..  (M2<oc) 

91.  log  u  =  (u  -  i)  -    *-  (u  -  i)2  +  -     (w  -  i)»  -   .  .  .  (2>  w>  o) 


93. 


94.    log  (i  +  M)  =  u  --  w2  H  --  «3  -  —  w4  +  . 

234 

95-      log  ferj)    =   2      tt  +  —  W3  +  —  M6  +  —  W7  +    .    .    .    1  (M2  <l) 

*>e^-,tt+ife>v7fe)<*;'^] 

M3  «5  M7 

97-    smh«  =  *  +  —  +—+—+...  («2<a) 

2 


[230] 


FORMULAS 

''  +JL--U 

6! 


98.      cOSh«=I+^-+-    -+-f  +    .    .    .  («2<cr) 


(0\  /  A    \  ,  »    » 

4«2\    /          4M-\   /      ,    4«2\ 
1       ~»*~J  V1       32r2/  V l  "*"  ?~W  '  *  *  («2  <  <*) 


99.     tanh  «  =  w «3H w8 —  w7  +  .  .  .  («2  <  i  T2) 

1,1  2 

100.  u  com  «  =  i  H «•* w4  H w8  —  .  fo2  O2) 

3  45  945 

1  •>  ,    5  6 1 

101 .  sech  u  =  i «^  +  —  «4 

2  24  720 

102.  U  CSCh  U  =   I T-U2  +  -~  U4  — 


108.    com"1 «  =  tanh"1  -  =  — I — -.  -\ r  H 7  +  .  . 

M       u      3  «*      5  M5       7  w7 

I          .         2  IM  13**  135 


6  360  15120 

103.  gdu  =  <t>  =  u  — z~w3H w5 «7  +  .  .  .  (M  small) 

r                         i  sech3  w      13  sech5  u 
= sech  u ...  (u  large) 

2  23  245 

I         -     .       I         s      .        6l          -  /  T\ 

104.  tt  =  ga1<£  =  <p-t--^<p'JH <p°H ^07+...  (0  <- 

6  24  5°40  \        2/ 

105.  sinh"1  «  =  «---+--—  -  -  -  4  —  +  .  .  .  («2<i) 

23        245       2467 

22  M2          244W4          2466  M6 

II          I3I          1351 

106.  cosh"1  u  =  log  2  w -9 — i -T  7—;  —  ...  («  >  i) 

22  W2          244  tt4          2466  tt6 
III  / 

107.  tanh"1  u  =  u  +  -  w  -\ —  «6  H —  w  +  .  .  .  («  <  i) 

3  5  7 


i  oo .    seen     w  *~  \^V/JI.A         —  AVCL 

M  «  22  244  2466 

II  II  I^I  1351  /  '\ 

no.     csch-'*  =  sinh"1-  ----_+--_-  ---  —,  +  .  .  . 

=  logf +I^_I3±4  +  I3|^_  B-<  (tt2<1) 

0  U        22          244         2466 


FORMULAS 


E.    DERIVATIVES 

deu 

in.    -7-  =  eu. 
du 

.  loge  U          I 

112.  d     j       =  -  • 

(III  U 

dav  dv    . 

113.  -r—  =  a",  -r-  .log, a. 
du  du 

114.  -3 —  =  uu  (i  +  loge «). 

J  sinh  M 

iis.    : =  coshtt. 

du 

d  cosh  M        .  , 

116.  •: =  smhtt. 

du 

d  tanh  u 

117.  —-j =  sech2«. 

d  coth  u 

118.    ; =  —  csch2  M. 

du 

d  sech  u 

no.    ; =—  sechw.  tanhM. 

du 

d  csch  u 

1 20.     — ; =  —  csch  u.  coth  u. 

du 


121 


d  sinh"1  u 


du  ^#2  _|_  j 

J  cosh"1  M  i 

122.        — = — 


123. 


du 
d  tanh"1  u 


d coth"1  u 

124.     -j = 

du 

d sech"1 u 

125.     ; =  — 

du  u 

126. 


du  u  ^  +  z 

dgd  u 

127.  —3 —  =  sechw. 

du 

d  gd-1 « 

128.  —^3 =  secw. 

du 

[232] 


FORMULAS 


F.    INTEGRALS.     (INTEGRATION  CONSTANTS  ARE  OMITTED.) 

1 29.  I  sinh  u  du  =  cosh  u. 

130.  I  cosh  udu  =  sinh  u. 

131.  I  tanh  udu  =  log  cosh  «. 

132.  I  coth  udu  =  log  sinh  «. 

X33-      /  sech  udu  =  2  tan"1  eu  =  gd  u. 

134.  I  csch  udu  =  log  tanh  -  - 

135.  /  sinh"  u  du  =  -  sinh  "-'  M.  cosh  u  —  — /  sinh  n~2  u  du, 

J  w  n     J 

I  w  _|_  2    /* 

=  — ; —  sinh  "4l  M  cosh  w I  sinh  n+2  M  rfw. 

»  +  i  w  +  i  J 

136.  I  cosh" udu  =  -  sinh w.  cosh n-1  M  + /  cosh n~? w dw, 

J  w  w     J 


sinh  u  coshn+1  u  -\ — ^  /  coshn+2 «  du. 


n  +  i 


137.  I  u  sinh  udu  =  u  cosh  M  —  sinh  M. 

138.  I  u  cosh  udu  =  u  sinh  w  —  cosh  «. 

139.  /  w2  sinh  udu  =  (u2  +  2)  cosh  u  —  2  u  sinh  «. 

140.  I  M"  sinh  udu  =  un cosh «  —  ww""1  sinh  u  +  n(n  —  i)  I  «n~s si 

141.  I  sinh2  w  JM  =  \  (sinh  M  cosh  u  —  u). 

142.  I  sinh  «.  cosh  u  du  =  \  cosh  (2  «). 
^ 

143.  /  cosh2  u  du  =  %  (sinh  u  cosh  «  +  M). 


[233] 


FORMULAS 

144.  I  tanh2  u  du  =  u  —  tanh  w. 

145.  I  coth2  u  du  =  u  —  coth  u. 

146.  I  sech2«rf«  =  tanhw. 

147.  I  sech3  M  du  =  i  sech  w  tanh  u  +  \  gd  u. 

148.  I  csch2  udu=—  coth  «. 

149.  /  sinh-1  u  du  =  u  sinh-1 «  —  (i  +  w2)*. 

150.  I  cosh-1  u  du  =  u  cosh"1  u  —  (w2  —  i)*. 

151.  /  tanh-1  udu  =  u  tanh-1  w  +  i  log  (i  —  w2). 

152.  T«  sinh-1  u  du  =  i     (2  w2  +  i)  sinh-1  u  -  u  (i  +  «2)*    . 
153-  TM  cosh-1  u  du  =  l\  (2  u2  -  i)  cosh-1  u  -  u  (u?  -  i)»    . 

154.  i  (cosh  a  +  cosh  w)"1  du  =  2  csch  o.  tanh"1  (tanh  ^  w.  tanh  £  a), 

=  csch  a     log  cosh  £  (w  +  a)  -  log  cosh  \(u  -  a)    . 

155.  I  (cos  a  +  cosh  u)~l  du  -  2  esc  a.  tan-1(  tanh  $  w.  tan  \  a). 

156.  /  (i  +  cos  a.  cosh  w)"1  du  =  2  esc  a.  tanh-1  (tanh  i  w.  tan  £  a). 

157.  I  sinh  «  cos  «•</«  =  i  (cosh  M.  cos  u  +  sinh  «.  sin  «). 

158.  /  cosh  «.  cos  udu=  \  (sinh  w.    cos  u  +  cosh  w.  sin  «). 

159.  I  sinh  w.  sin  M  rfw  =  5  (cosh  u.  sin  «  —  sinh  u.  cos  w). 

160.  I  cosh  w.  sin  M  du  =  i  (sinh  w.  sin  u  —  cosh  «.  cos  «). 


[234] 


FORMULAS 


161.  I  sinh  (mu)  sinh  (nu)  du 

— -2     m  sinh  (nu)  cosh  (mu)  —  n  cosh  (nu)  sinh  (w«)    . 
m  -    n    L  J 

162.  I  cosh  (WM)  sinh  (nu)  du 

m  sinh  (nu)  sinh  (w«)  —  w  cosh  (nu)  cosh  (w«) 
w        n    L  J 

163.  I  cosh  (WM)  cosh  («M)  rfw 

=  — 5     m  sinh  (mu)  cosh  (WM)  —  n  sinh  (««)  cosh  (mu)    . 

m  —  n    \_ 

164.  /  sinh  u  tanh  u  du  =  sinh  u  —  gd  u. 

/u 
cosh  u  coth  udu  =  cosh  M  +  log  tanh  —  • 

1 66.  I  sec  M  du  —  gd"1 «. 

167.  I  sec3  <j>d<t>  =  I  (i  +  tan2  </>)*  J  tan  <£  =  £  sec  <£  tan  0  +  |  gd-1 0, 

=  |  tan  $  (i  +  tan2  <£)*  +  \  sinh-1  (tan  0).    Here  <f>  =  gd  u. 


r      du  .u  r      du 

168.      I     ,  „   ,     ,..     =  sinh-1--  f  -p-j  -  5T7—  =  sin"1 

J     (u2  +  a2)*  a  J     (a2  —  « 


n 
')*     """     a' 


r        du                       ,u  r      —  du                  .u 

l6o.        I    —  7-S  -  ^T  =  COSh"1   -    •  I  —r-~  --  5rr-  =  COS    1—  • 

J      (u2  —  a2)*                   a  J     (a2  —  «2)*                  a 

/<f  w               i           .  u  r        du    *           i  A       .  « 

7-5  --  5x  —  =  -  tanh"1  -  •  ,  ,     ,  —  =  -  tan-1  -  • 

(a2  -  w2)u<o      a              «  J        a2  +  w2           a            a 

/—  J  M             i            ,«  C       —  du             i          .« 

r~5—      v  -  =  -  coth-1  —  /  -  r^—i  —  =  ~  cot    -* 

(w2-a2)u>a      a             a  J       a2  +  «2           a           a 

/—  d«             i           .  «  f         du               i            u 

—  —  -  —  -  =  -  sech-1  -  •  I  —  7-5  -  ^T-  =  —  sec  *  -  • 

M  (a2  —  «2)*       a             a  J  u  (u2  —  a2)*       a           a 

/—  d«            i           ,  u  f       —  du            i           u 

u(a*  +  u2)*  =aCSCh    a"  J   «  («2  -  a2)   =  a  C!       a 

/du                       i    ,  aw  +  b 

.     ,   ,              —  TT  =     =  sinh-1  7  --  rjrr-  ,  a  positive,  ac  >  b2; 

(au2  +  2bu  +  c)*        v«  C«  -  P)1 


i  ,    au  +  &  ...  ^  ,, 

=  -=  cosh"1  ^75— — y  ,  a  positive,  ac  <  b2; 

=  — —  cos"1  ,  ^U       X1>  a  negative. 

(b2  —  ac)» 

[235] 


FORMULAS 


/ 


du 


(au2 


tan 


au  +  b 


(ac  -  62)*  (ac  - 

—  i       •"  v-i    au  +  b 


(b*  -  a, 


tanlr 


(b2  -  ac)* 


ac  >  b*; 

ac  <  b2, 
au  +  b  <  (b2  —  ac)*. 

ac  <  62, 

au  +  6  >  (62  -  ac)*. 


176. 


J  (a- 


u)  (u  -  6)*       (a  -  6)* 


—  b 


a  —  b 


or 


or 


—  2  \u  —  b 

tan-1 


(b  -  a)* 

2 


(a  -  6)i 


coth-1 


I77< 


f 

J  (a- 


du 


b~=~a' 

u  —  b 
a  —  b 

tanh"1 


u)  (b  -  w)*       (b  -  a)* 

2  \b  —  u 


or 


or 


TT  -  rr 
(b  —  a)* 

-.  -  TTJ 
(a  —  b)* 


coth-1 


tan 


a 

-  - 
a  —  b 


(The  real  form  is  to  be  taken.) 

b  —  u 
b  —  a 

(The  real  form  is  to  be  taken.) 


178. 
179. 
1 80. 
181. 

182. 

183- 
184. 
185. 


=  i  w  («2  —  a2)*  —  |  a2  cosh"1  — 

a 


«2)*  JM  =  ^  u  (a2  -  «2)i  +  ^  a2  sin"1  - 

o> 


du= 


a2)* 


/<«'- 

JV- 

JV  + 

//flu 
**  u~~a'' 

/gOU 
utf*u  du  =  -j  (aw  —  i). 
a 

/Um     MU                   yy>          f* 
um-(Pu  du  = I  um~l  e?u  du. 
a           aj 

/e?u  du  _       i  e°"  /^e°u  a7!*  1 

um         m  —  i  [      « ""^        «/    w*""1  J 

//T&« 
a^rfw  = 


6  log  a 


[236] 


FORMULAS 

C  _auun  _  nau  un~l      n  (n  —  i)  au  un  ~2 

J  log  a        (log  a)2  (log  a)3 

n  (n  —  i)  (n  —  2)    •   •    2.  i  a" 
(log  a)n+1 

l8?   ra"^M  =  a"  r E iqgg    _       (log  a)2 

J     un         n  —  i  [       «  "-1       (w  —  2)  w""2       (w  —  2)  (w  -  3)  «»-; 

(log  a)  "-1 ra"du  1 

(w-  2)  (w-  3)    ....  2.iJ  ~u~ \' 

Cau  du  _  .  (u  log  a)2       (« log  a)3 

J        U  2.2  I  3.3  ! 

du  eu 


du 


-  log  (  V  a  +  bemu  +  V"a)    . 

/ueu  du  e" 

(i  +  «)2   ~TT«  " 

/e°ulog« 
» 


195.  I  logudu  =  u  log  u  —  u. 

196.  Jumlogudu  =  um+l  [^^  ~  (w+l)2J- 

197.  I  (log  «)"  du  =  u  (log  w)n  —  n  I  (log  w)"-1  rfw. 

r       /i  w>n+1  (l°g M)"  n       C  m  n        ^n- 

198.  /  «m  (log  u)ndu  =  -  I  wm  (log  «)" 

«y  m  +  i  w  -f-  i«7 


20°- 


(log  «)n  </M  _  (log  u)n+l  ^ 
u  n  +  i 

dw  ,   (logw)2  .   (log«) 


[237] 


FORMULAS 
r     du  u 

2OI 


2O2 


/du  u  i       r      du 

(log «)n  ~       (w  -  i)  (log tt)"-1      «  -  ij  (logtt)"-1' 

/MTO  o#                       um+l                  m  +  i   /*    um  du 
.,  i  *^~  —   -4—    '  I    * 

(log  «)n          (»  —  i)  (log  tt)""1       n  —  i  J  (log  «)n~ 

/«m  a"«        /^e~tf 
—  =   I  —  Jy,  where  y  =  —  (m  -f  i)  log  w. 
log  u      J    y 

/du 
—  :=  log  (log  u). 
u  log  u 

/du                              i 
_  —  . 
u  (log  u)n          (n  —  i)  (log  w)n~ 

206.  I  (a  +  bu)m  log  udu  = 

— r     (a  +  to)*+1log  u  —  I  — 
o  (w  +  i)  L  J  w  J 

207.  I  «m  log  (a  +  to)  O"M  = 

- —     u"*1  log  (fl  +  to)  -  6  /  — 
w+iL  J    a  +  bu  \ 

208.  I  — -    = 

J  u 

bu 


/logM^M       _  I  logW  /*  </ 

(a  +  to)*  "  6  (f»  -  i)  L  ~  (a  +  to)"1-1  +  J  w(a  + 

/log udu        i    '          ,      /     ,   T  N       i    Tlog  (a  +  to) 
^-^==_ 

/,  (a  +  to)2 . 

(a  +  bu)  log  u  du  =  -  -  log  « 

20 


f(a  +  bu)2 .              a2  log  u 
(a  +  bu)  log  u  du  =  -    — ; —  -  log  u ^ au  — 


fk 

212.        I   7- 

J  (a 


log  M  C?M 

+  to)*  ~ 


(log  u  —  2)  V(«  +  to)  +  V  a  log  ( Va  +  to  +  V  a) 

*  L 

-  V  a  log  ( V  a,  4-  to  —  V  a)     ,  if  a>  o, 
=  -T-   (log  M  —  2)  V  (&  +  bu)  +  2  V  —  a  tan"1  -vl  -      — ,  if  a  <  o. 


[238] 


FORMULAS 

213.  Ce~<**du  =  -^  =—  r(i). 
Jo  2  a         20, 

214.  ru»e-*«du=T(n+J  =  ^ 
Jo  an+l         an 

«s.  r»-r~  ^=i 

Jo 


- 
+1 


. 

2  n    a 


2i6.      I    e  "*~  ^r  du  =  -    -  \  IT    •  a  >  o. 


.      I    e~"*~  &  du  =  -  A|  IT 
Jo  2      > 

/*"  i        [TT 

217.       '    e"""  V  M  d«  =  --  A  -  ' 
Jo  2n    V  n 

/*e-»u  17 

==~du  =  \  —  • 
i  Vw  >  w 

f00        ^M  7T 

219.  .    ,     ,  -  r  =   -   • 

J  o  sinh  (w«)       2  w 

f00      M^M  7T2 

.  .    ,    (  -  r-  =  -  5- 

J  o  smh  (nu)        4  wj 


220 


221 


f*it  /»IT 

sinh  (mu) .  sinh  (««)  du  =  I      cosh  (WM)  .  cosh  (»«)  < 
*/  o  Jo 

=  o,  if  m  is  different  from  n. 

/it  (*i*  i^ 

cosh2  (mu)  du  =—  I      sinh2  (w«)  </w  =  —  • 
i  Jo  2 

£it 
sinh  (WM)  JM  =  o. 
* 

rir 

224.      I      cosh  (mu)  du  =  o. 
J  o 

/tf 
sinh  (w«)  cosh  (nu)  du  =  o. 
-it 

pit 

226.  I      sinh  (mu)  cosh  (ww)  </M  =  o. 
J  o 

T1    logM     ,  7T2 

227.  -  du  = * 

Jo  i  -  «  6 

/"!    log  U  IT2 

228.  I  .  — - —  du  = 

Jo    I   +  U  12 

Cl  log  u   , 

229.  '—  du  =  — 
Jo  i-u2 


*. 

8 

[239] 


FORMULAS 


r1  .  /  I   +  U  \        du          7T* 

230.  log  {  )  •  —  =  — 

Jo      3  \i  -  u)      u        4 


du 


-u«)du 


233.     /    (logw)n<fo=  (- 

J  0 


r/.  i\» 

Jo  (log-j 


237. 


[240] 


1 MTVERSITY  of  CALIFORNIA 

AT 

LOS  ANGELES 
LIBRARY 


Engmeering  & 
Mathematical 

Sciences 

Library 


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